Research Article Certain Families of Multivalent Analytic...
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Research Article Certain Families of Multivalent Analytic Functions Associated with Iterations of the Owa-Srivastava Fractional Differintegral Operator A. K. Mishra 1 and S. N. Kund 2 1 National Institute of Science and Technology, Palur Hills, Berhampur 761008, India 2 Department of Mathematics, Khallikote Autonomous College, Ganjam District, Berhampur, Odisha 760001, India Correspondence should be addressed to A. K. Mishra; [email protected] Received 19 May 2014; Accepted 18 September 2014; Published 14 October 2014 Academic Editor: Haakan Hedenmalm Copyright © 2014 A. K. Mishra and S. N. Kund. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By making use of a multivalent analogue of the Owa-Srivastava fractional differintegral operator and its iterations, certain new families of analytic functions are introduced. Several interesting properties of these function classes, such as convolution theorems, inclusion theorems, and class-preserving transforms, are studied. 1. Introduction Let A denote the class of analytic functions in the open unit disk U = { : ∈ C, || < 1} (1) and let A be the subclass of A consisting of functions represented by the following Taylor-Maclaurin’s series: () = + ∞ ∑ =1 + ( ∈ N := {1, 2, 3, . . .}) . (2) In a recent paper Patel and Mishra [1] studied several interesting mapping properties of the fractional differintegral operator: Ω (,) : A → A ( ∈ N, −∞ < < + 1, ∈ U), (3) defined by Ω (,) () := + ∞ ∑ =1 Γ ( + + 1) Γ ( + 1 − ) Γ ( + 1) Γ ( + + 1 − ) + ( ∈ U), (4) where ∈ A is given by (2). In the particular case =1 and −∞ < < 1 the fractional-differintegral operator Ω (,1) := Ω (5) was earlier introduced by Owa and Srivastava [2] (see also [3]) and this is popularly known as the Owa-Srivastava operator [4–6]. Moreover, for 0≤<1 and ∈ N, Ω (,) was investigated by Srivastava and Aouf [7] which was further extended to the range −∞<<1, ∈ N by Srivastava and Mishra [8]. e following are some of the interesting particular cases of Ω (,) : Ω (−1,) () = +1 ∫ 0 () , Ω (0,) () = () , Ω (1,) () = () , Ω (,) () = ( − )! () ! , ( ∈ N; < + 1) , Ω (−,) () = + ∫ 0 −1 Ω (−+1,) () ( ∈ N). (6) Hindawi Publishing Corporation Journal of Complex Analysis Volume 2014, Article ID 915385, 9 pages http://dx.doi.org/10.1155/2014/915385
Transcript of Research Article Certain Families of Multivalent Analytic...
Research ArticleCertain Families of Multivalent Analytic FunctionsAssociated with Iterations of the Owa-Srivastava FractionalDifferintegral Operator
A K Mishra1 and S N Kund2
1 National Institute of Science and Technology Palur Hills Berhampur 761008 India2Department of Mathematics Khallikote Autonomous College Ganjam District Berhampur Odisha 760001 India
Correspondence should be addressed to A K Mishra akshayam2001yahoocoin
Received 19 May 2014 Accepted 18 September 2014 Published 14 October 2014
Academic Editor Haakan Hedenmalm
Copyright copy 2014 A K Mishra and S N KundThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
By making use of a multivalent analogue of the Owa-Srivastava fractional differintegral operator and its iterations certain newfamilies of analytic functions are introduced Several interesting properties of these function classes such as convolution theoremsinclusion theorems and class-preserving transforms are studied
1 Introduction
LetA denote the class of analytic functions in the open unitdisk
U = 119911 119911 isin C |119911| lt 1 (1)
and let A119901 be the subclass of A consisting of functionsrepresented by the following Taylor-Maclaurinrsquos series
119891 (119911) = 119911
119901+
infin
sum
119896=1
119886119896119911119896+119901
(119901 isin N = 1 2 3 ) (2)
In a recent paper Patel and Mishra [1] studied severalinteresting mapping properties of the fractional differintegraloperator
Ω
(120582119901)
119911 A119901 997888rarr A119901 (119901 isin N minusinfin lt 120582 lt 119901 + 1 119911 isin U)
(3)
defined by
Ω
(120582119901)
119911119891 (119911) = 119911
119901
+
infin
sum
119896=1
Γ (119896 + 119901 + 1) Γ (119901 + 1 minus 120582)
Γ (119901 + 1) Γ (119896 + 119901 + 1 minus 120582)
119886119896119911119896+119901
(119911 isin U)
(4)
where 119891 isin A119901 is given by (2) In the particular case 119901 = 1
and minusinfin lt 120582 lt 1 the fractional-differintegral operator
Ω
(1205821)
119911= Ω
120582
119911(5)
was earlier introduced byOwa and Srivastava [2] (see also [3])and this is popularly known as the Owa-Srivastava operator[4ndash6] Moreover for 0 le 120582 lt 1 and 119901 isin N Ω(120582119901)
119911was
investigated by Srivastava and Aouf [7] which was furtherextended to the range minusinfin lt 120582 lt 1 119901 isin N by Srivastavaand Mishra [8] The following are some of the interestingparticular cases of Ω(120582119901)
119911
Ω
(minus1119901)
119911119891 (119911) =
119901 + 1
119911
int
119911
0
119891 (120585) 119889120585
Ω
(0119901)
119911119891 (119911) = 119891 (119911)
Ω
(1119901)
119911119891 (119911) =
119911119891
1015840(119911)
119901
Ω
(119899119901)
119911119891 (119911) =
(119901 minus 119899)119911
119899119891
119899(119911)
119901
(119899 isin N 119899 lt 119901 + 1)
Ω
(minus119898119901)
119911119891 (119911) =
119901 + 119898
119911
119901int
119911
0
119905
119898minus1Ω
(minus119898+1119901)
119911119891 (119905) 119889119905 (119898 isin N)
(6)
Hindawi Publishing CorporationJournal of Complex AnalysisVolume 2014 Article ID 915385 9 pageshttpdxdoiorg1011552014915385
2 Journal of Complex Analysis
Furthermore
119911(Ω
(120582119901)
119911119891 (119911))
1015840
= (119901 minus 120582)Ω
(120582+1119901)
119911119891 (119911)
+ 120582Ω
(120582119901)
119911119891 (119911) (minusinfin lt 120582 lt 119901 119911 isin U)
(7)
The 119899-iterates of the operatorΩ(120582119901)119911
are defined as follows
Ω
(120582119901)
1199110119891 (119911) = 119891 (119911) (8)
and for 119899 isin N
Ω
(120582119901)
119911119899119891 (119911) = Ω
(120582119901)
119911(Ω
(120582119901)
119911119899minus1119891 (119911))
= 119911
119901+
infin
sum
119896=1
[
Γ(119896 + 119901 + 1)Γ(119901 + 1 minus 120582)
Γ(119901 + 1)Γ(119896 + 119901 + 1 minus 120582)
]
119899
119886119896119911119896+119901
(9)
Similarly for 119891 isin A119901 represented by (2) let the operator
C119898
119905 A119901 997888rarr A119901 (119898 isin N 119905 isin C) (10)
be defined by the following
C0
119905119891 (119911) = 119891 (119911)
C1
119905119891 (119911) = (1 minus 119905) 119891 (119911) +
119905
119901
(119911119891
1015840(119911))
= 119911
119901+
infin
sum
119896=1
1 +
119905119896
119901
119886119896119911119896+119901
(11)
and for119898 isin N 1
C119898
119905119891 (119911) = C
1
119905(C119898minus1
119905119891 (119911)) = 119911
119901+
infin
sum
119896=1
1 +
119905119896
119901
119898
119886119896119911119896+119901
(12)
Very recently Srivastava et al [6] considered the compositionof the operatorsC119898
119905and Ω(120582119901)
119911119899and introduced the following
operator
D(120582119901)
119905 (119899119898) A119901 997888rarr A119901 (119899119898 isin N 1 119905 isin C) (13)
That is for 119891 isin A119901 given by (2) we know that
D(120582119901)
119905 (119899119898) 119891 (119911)
= Ω
(120582119901)
119911119899(C119898
119905119891 (119911))
= C119898
119905(Ω
(120582119901)
119911119899119891 (119911))
= 119911
119901+
infin
sum
119896=1
[
Γ (119896 + 119901 + 1) Γ (119901 + 1 minus 120582)
Γ (119901 + 1) Γ (119896 + 119901 + 1 minus 120582)
]
119899
times 1 +
119905119896
119901
119898
119886119896119911119896+119901
(119899119898 isin N0 = N cup 0 minusinfin lt 120582 lt 119901 + 1 119905 ≧ 0 119911 isin U)
(14)
The transformation D(120582119901)
119905 (119899119898) includes among many thefollowing two previously studied interesting operators asparticular cases
(i) For 119899 = 119898 isin N 119901 = 1 0 le 120582 lt 1 the frac-tional derivative operator D(1205821)
119905(119899 119899) = D119899120582
119905was
recently introduced and investigated by Al-Oboudiand Al-Amoudi [9 10] in the context of functionsrepresented by conical domains
(ii) For 119899 = 119898 isin N 119901 = 1 120582 = 0 119905 = 1 D(01)1
(119899 119899) is the Salagean operator [11] which is in factthe 119899-iterates of the popular Alexanderrsquos differentialtransform 119891(119911) rarr 119911119891
1015840(119911) [12]
We next recall the definition of subordination Supposethat 119891 isin A and 119892 inA is univalent in U We say that 119891(119911) issubordinate to 119892(119911) in U if 119891(0) = 119892(0) and 119891(U) sube 119892(U)Considering the function 119908(119911) = 119892
minus1(119891(119911)) it is readily
checked that 119908(119911) satisfies the conditions of the Schwarzlemma and
119891 (119911) = 119892 (119908 (119911)) (119911 isin U) (15)
In a broader sense the function 119891 isin A is said to be sub-ordinate to the function 119892 isin A (119892 need not be univalent inU) written as
119891 ≺ 119892 in U or 119891 (119911) ≺ 119892 (119911) (119911 isin U) (16)
if condition (15) holds for some Schwarz function 119908(119911) (see[12] for details) We also need the following definition ofHadamard product (or convolution) For the functions 119891 and119892 inA119901 given by the following Taylor-Maclaurinrsquos series
119891 (119911) = 119911
119901+
infin
sum
119896=1
119886119896119911119896+119901
119892 (119911) = 119911
119901+
infin
sum
119896=1
119887119896119911119896+119901
(119911 isin U)
(17)
their Hadamard product (or convolution) 119891 lowast 119892 is defined by
(119891 lowast 119892) (119911) = 119911
119901+
infin
sum
119896=1
119886119896119887119896119911119896+119901
(119911 isin U) (18)
It is easy to see that 119891 lowast 119892 isin A119901The study of iterations of entire and meromorphic func-
tions as the number of iterations tends to infinity is a populartopic in complex analysis However investigations havebeen initiated only recently regarding iterations of certaintransforms defined on classes of analytic and meromorphicfunctions For example Al-Oboudi and Al-Amoudi [9 10]investigated properties of certain classes of analytic functionsassociated with conical domains by making use of theoperator D119899120582
119905 Their work generalized several earlier results
of Srivastava and Mishra [13] This theme has been furtherpursued in our more recent papers [6 14 15] In the sequel to
Journal of Complex Analysis 3
these current investigations in the present paper we definethe following subclass of A119901 associated with the iteratedoperator D(120582119901)119905 (119899119898) and investigate its several interestingproperties Our work is also motivated by earlier works in[16ndash20] connecting subordination and Hadamard product
Definition 1 The function 119891 isin A119901 is said to be in theclassH119899119898
119901(120582 119905 ℎ) if the following subordination condition is
satisfied
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
≺ ℎ (119911)
(119911 isin U)
(19)
where 119905 is a complex number and ℎ is an analytic convexunivalent function in U
The function class H119899119898119901(120582 119905 ℎ) includes several previ-
ously studied subclasses ofA as particular cases For example
(i) for 119899 = 1 119898 = 0 the classH10119901(120582 119905 ℎ) = H119901(120582 119905 ℎ)
was recently studied by Liu [17](ii) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 119905 = 1 and ℎ(119911) =
(1 + 119911)(1 minus 119911) the classH101(1 1 (1 + 119911)(1 minus 119911)) was
earlier investigated by R Singh and S Singh [21](iii) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 and ℎ(119911) = (1 +
119886119911)(1+119887119911) (minus1 le 119887 lt 119886 le 1) H101(1 119905 (1+119886119911)(1+
119887119911)) reduces to H(119905 119886 119887) which was investigated byYang [22]
(iv) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 and ℎ(119911) =
1+119872119911 (119872 gt 0) H101(1 119905 1+119872119911) reduces toS(119905119872)
the class introduced and studied by Zhongzhu andOwa [23] and Jinlin [24]
In the present paper we primarily focus on a variety ofconvolution theorems for the class H119899119898
119901(120582 119905 ℎ) We also
find inclusion theorems and study behavior of the Libera-Livingston integral operator
2 Some More Definitions andPreliminary Lemmas
We need the following definitions and results for the presen-tation of our results Let CV(120588) and Slowast(120588) (0 le 120588 lt 1)
denote respectively the classes of univalent convex functionsof order 120588 and starlike functions of order 120588 (see [12] for details)The function 119891 isin A1 is said to be in the class PSlowast(120588)consisting of prestarlike functions of order 120588 [25] if
119911
(1 minus 119911)
2(1minus120588)lowast 119891 (119911) isin S
lowast(120588) (20)
It is readily seen that
PSlowast(0) = CV (0) = CV PS
lowast(
1
2
) = Slowast(
1
2
)
(21)
Furthermore it is well known [25] that
PSlowast(120583) sub PS
lowast(120582) (120583 le 120582) (22)
Wewill also need the following lemmas in order to derive ourmain results
Lemma 2 (see [26] also see [27]) Let 119892 be analytic in U andlet ℎ be analytic and convex univalent in U with ℎ(0) = 119892(0)If
119892 (119911) +
1
120583
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U) (23)
whereR(120583) ge 0 and 120583 = 0 then
119892 (119911) ≺ℎ (119911) = 120583119911
minus120583int
119911
0
120585
120583minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(24)
and ℎ is the best dominant of (24)
Lemma 3 (see [25]) Let 120588 lt 1 119891 isin Slowast(120588) and 119892 isin PSlowast(120588)Then for any analytic functionF in U
119892 lowast (119891F)
119892 lowast 119891
(U) sub 119862119874 (F (U)) (25)
where 119862119874(F(U)) denotes the closed convex hull ofF(U)
Lemma 4 (see [28]) Let 119867 and 119866 be univalent convexfunctions inU and let ℎ and 119892 be functions inA Suppose thatℎ ≺ 119867 and 119892 ≺ 119866 in U Then ℎ lowast 119892 ≺ 119867 lowast 119866 in U
The following well known result is a consequence of theprinciple of subordination and can be found for example in[12 29]
Lemma 5 Let the function 119892 isin A satisfy 119892(0) = 1 andR(119892(119911)) gt 1205730 (0 le 1205730 lt 1 119911 isin U) Then
R (119892 (119911)) gt 1205730 + (1 minus 1205730)1 minus |119911|
1 + |119911|
(119911 isin U) (26)
3 Convolution Results
We state and prove the following convolution results
Theorem 6 Let 119891 isin H119899119898119901(120582 119905 ℎ) and suppose that 119892 in A119901
satisfies the following
R 119911
minus119901119892 (119911) gt
1
2
(119911 isin U) (27)
Then
119891 lowast 119892 isin H119899119898
119901(120582 119905 ℎ) (28)
4 Journal of Complex Analysis
Proof For every 119891 and 119892 inA119901 we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
= (1 minus 119905) (119911
minus119901119892 (119911)) lowast (119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911))
+
119905
119901
(119911
minus119901119892 (119911)) lowast (119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
)
= (119911
minus119901119892 (119911)) lowast Ψ (119911)
(29)
where
Ψ (119911) = (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
(30)
Now if 119891 isin H119899119898119901(120582 119905 ℎ) then
Ψ (119911) ≺ ℎ (119911) (119911 isin U) (31)
Furthermore condition (27) is equivalent to
119911
minus119901119892 (119911) ≺
1
1 minus 119911
(119911 isin U) (32)
Therefore an application of Lemma 4 in (29) yields
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
≺
1
1 minus 119911
lowast ℎ (119911)
= ℎ (119911)
(33)
This shows that 119891 lowast 119892 isin H119899119898119901(120582 119905 ℎ) The proof of
Theorem 6 is completed
Remark 7 Taking 119899 = 1 119898 = 0 inTheorem 6 we get a recentresult of Liu ([17]Theorem 3)The choices 119899 = 1 119898 = 0 119901 =1 120582 = 1 and ℎ(119911) = (1 + 119886119911)(1 + 119887119911) (minus1 le 119887 lt 1 119886 gt 119887)
yield a result of Yang ([22] Theorem 4)
Corollary 8 Let the function 119891 given by (2) be a member ofH119899119898119901(120582 119905 ℎ) and
119904119903 (119911) = 119911119901+
119903minus1
sum
119896=1
119886119896119911119896+119901
(119903 isin N 1 119911 isin U) (34)
Then the function
120590119903 (119911) = int
1
0
119905
minus119901119904119903 (119905119911) 119889119905
(35)
is in the classH119899119898119901(120582 119905 ℎ)
Proof Let 119891 isin H119899119898119901(120582 119905 ℎ) We note that
120590119903 (119911) = 119911119901+
119903minus1
sum
119896=1
119886119896
119896 + 1
119911
119896+119901= (119891 lowast 119892119903) (119911) (119903 isin N 1)
(36)
where
119891 (119911) = 119911
119901+
infin
sum
119896=1
119886119896119911119896+119901
119892119903 (119911) = 119911
119901+
119903minus1
sum
119896=1
119911
119896+119901
119896 + 1
isin A119901
(37)
Also for 119903 isin N 1 it is well known [18] that
R 119911
minus119901119892119903 (119911) gt
1
2
(119911 isin U) (38)
In view of (36) and (38) an application of Theorem 6 gives
120590119903 isin H119899119898
119901(120582 119905 ℎ) (39)
The proof of Corollary 8 is completed
Theorem 9 Let the function 119892 in A119901 be such that 119911minus119901+1119892(119911)is a prestarlike function of order 120588 (120588 lt 1) If119891 isin H119899119898
119901(120582 119905 ℎ)
then119891 lowast 119892 isin H
119899119898
119901(120582 119905 ℎ) (40)
Proof Let 119891 isin H119899119898119901(120582 119905 ℎ) and 119892 isin A119901 Then (29) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
=
(119911
minus119901+1119892 (119911)) lowast (119911Ψ (119911))
(119911
minus119901+1119892 (119911)) lowast 119911
(119911 isin U)
(41)
where Ψ(119911) is defined as in (30) We noted in the proof ofTheorem 6 thatΨ(119911) ≺ ℎ(119911) Since 119911minus119901+1119892(119911) isin PSlowast(120588) 119911 isinSlowast(120588) and ℎ(119911) is convex univalent in U an application ofLemma 3 in (41) yields the following
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)(119891 lowast 119892)(119911))
1015840
≺ ℎ (119911) (119911 isin U)
(42)
Therefore 119891 lowast 119892 isin H119899119898119901(120582 119905 ℎ) The proof Theorem 9 is
completed
Taking 120588 = 12 in Theorem 9 we get the following
Corollary 10 Let 119891 isin H119899119898119901(120582 119905 ℎ) and suppose that 119892 inA119901
is such that 119911minus119901+1119892 isin Slowast(12) Then
119891 lowast 119892 isin H119899119898
119901(120582 119905 ℎ) (43)
In particular if 119911minus119901+1119892 is univalent convex then 119891 lowast 119892 isin
H119899119898119901(120582 119905 ℎ)
Journal of Complex Analysis 5
In the following theorem we discuss convolution proper-ties of the function classH(120582 119905 ℎ)when ℎ is a right half planemapping
Theorem 11 Let 119905 ge 0 and suppose that each of the functions119891119895 (119895 = 1 2) is a member of the classH119899119898
119901(120582 119905 ℎ119895) where
ℎ119895 (119911) =
1 + (1 minus 2120573119895) 119911
1 minus 119911
(0 le 120573119895 lt 1) (44)
If 119891 isin A119901 is defined by the following
119891 (119911) = D(120582119901)
119905 (119899119898) (1198911 lowast 1198912) (119911) (119911 isin U) (45)
then 119891 isin H119899119898119901(120582 119905 ℎ) where
ℎ (119911) =
1 + (1 minus 2120573) 119911
1 minus 119911
(46)
and 120573 is given by
120573 =
1 minus 4 (1 minus 1205731) (1 minus 1205732)
times(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906) (120582 gt 0)
1 minus 2 (1 minus 1205731) (1 minus 1205732) (120582 = 0)
(47)
The bound on 120573 is the best possible
Proof We consider the case 119905 gt 0 Suppose that 119891119895 isin
H119899119898119901(120582 119905 ℎ119895) (119895 = 1 2) where ℎ119895(119911) is given by (44) By
setting
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891119895(119911))
1015840(48)
we see in the light of Definition 1 that
Φ119895 (119911) ≺ ℎ119895 (119911) (119895 = 1 2) (49)
A routine calculation yields the following
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1Φ119895 (120585) 119889120585
(119895 = 1 2)
(50)
Now if 119891(119911) is defined by (45) then using (50) we get that
D(120582119901)
119905 (119899119898) 119891 (119911)
= D(120582119901)
119905 (119899119898) 1198911 (119911) lowastD(120582119901)
119905 (119899119898) 1198912 (119911)
= (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ1 (119906119911) 119889119906)
lowast (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ2 (119906119911) 119889119906)
=
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ0 (119906119911) 119889119906
(51)
where
Φ0 (119911) =
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1(Φ1 lowast Φ2) (119906119911) 119889119906
(52)
Since
Φ1 (119911) minus 1205731
1 minus 1205731
≺
1 + 119911
1 minus 119911
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
≺
1
1 minus 119911
(53)
by using Lemma 4 we get
(
Φ1 (119911) minus 1205731
1 minus 1205731
) lowast (
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
) ≺
1 + 119911
1 minus 119911
(54)
A simple calculation gives that
Φ1 (119911) lowast Φ2 (119911) ≺ 1 plusmn 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(55)
Therefore by the Lindeloff principle of subordinationwehave
R Φ1 (119911) lowast Φ2 (119911)
gt 1205730 = 1 minus 2 (1 minus 1205731) (1 minus 1205732) (119911 isin U) (56)
Since
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= Φ0 (119911)
(57)
by using (52) in conjunction with (56) and Lemma 4 we getthe following
R (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= R Φ0 (119911)
=
119901
119905
int
1
0
119906
(119901119905)minus1R (Φ1 lowast Φ2) (119906119911) 119889119906
ge
119901
119905
int
1
0
119906
(119901119905)minus1(1205730 + (1 minus 1205730)
1 minus 119906 |119911|
1 + 119906 |119911|
) 119889119906
gt 1205730 +
119901 (1 minus 1205730)
119905
int
1
0
119906
(119901119905)minus1 1 minus 119906
1 + 119906
119889119906
= 1 minus 4 (1 minus 1205731) (1 minus 1205732) (1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
= 120573 (119911 isin U)
(58)
This proves that119891(119911) isin H119899119898119901(120582 119905 ℎ) where the function ℎ(119911)
is given by (46)
6 Journal of Complex Analysis
In order to show that the value of 120573 is the least possiblewe take the functions 119891119895(119911) isin A119901 (119895 = 1 2) defined by
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1
times (120573119895 + (1 minus 120573119895)1 + 120585
1 minus 120585
) 119889120585
(59)
for which we have
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891119895 (119911))
1015840
= 120573119895 + (1 minus 120573119895)1 + 119911
1 minus 119911
(119895 = 1 2)
(60)
(Φ1 lowast Φ2) (119911) = 1 + 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(61)
Hence for 119891 isin A119901 given by (45) we obtain
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
=
119901
119905
int
1
0
119906
(119901119905)minus1(1 + 4 (1 minus 1205731) (1 minus 1205732)
119906119911
1 minus 119906119911
) 119889119906
997888rarr 120573 (as 119911 997888rarr minus1)
(62)
Finally for the case 119905 = 0 the proof of Theorem 11 is simpleso we choose to omit the details involved
4 Properties of the Libera-LivingstonTransform
For the function 119891 isin A119901 the functionF defined by
F (119911) =
120583 + 119901
119911
120583int
119911
0
120585
120583minus1119891 (120585) 119889120585 (R (120583) gt minus119901 119911 isin U)
(63)
is popularly known as the Libera-Livingston transform of 119891We state and prove the following
Theorem 12 Let 119891 isin H119899119898119901(120582 119905 ℎ) Then the function F
defined by (63) is in the classH119899119898119901(120582 119905
ℎ) where
ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
120585
120583+119901minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(64)
ConsequentlyF isin H119899119898119901(120582 119905 ℎ) The function ℎ(119911) is the best
dominant in (64)
Proof We define the function119870 on U by
119911
119901119870 (119911) = (1 minus 119905)D
(120582119901)
119905 (119899119898)F (119911)
+
119905
119901
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
(65)
Differentiating both the sides of (65) with respect to 119911 we get
119901119870 (119911) + 119911119870
1015840(119911)
= 119901 (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)
+ 119905119911
minus119901+1(D(120582119901)
119905 (119899119898)
119911F1015840(119911)
119901
)
1015840
(66)
Also the defining relation (63) yields
(120583 + 119901) 119891 (119911) = 120583F (119911) + 119911F1015840(119911) (67)
Now a routine calculation using (65) (66) and (67) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
(119911 isin U)
(68)
Since 119891 isin H119899119898119901(120582 119905 ℎ) we get the following from the
preceding equation (68)
119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
≺ ℎ (119911) (69)
Therefore by applying Lemma 2 we have
119870 (119911) ≺ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
119905
120583+119901minus1ℎ (119905) 119889119905 ≺ ℎ (119911)
(70)
This last subordination (70) is equivalent to
F isin H119899119898
119901(120582 119905
ℎ) sub H
119899119898
119901(120582 119905 ℎ) (71)
The proof of Theorem 12 is completed
Theorem 13 Let 119891 isin A119901 and suppose that the function F isdefined as in (63) If
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (120574 gt 0)
(72)
thenF isin H119899119898119901(120582 0
ℎ) where
ℎ (119911) =
120583 + 119901
120574
119911
minus(120583+119901)120574
times int
119911
0
120585
((120583+119901)120574)minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (R (120583) gt minus119901)
(73)
The function ℎ(119911) is the best dominant in (73)
Journal of Complex Analysis 7
Proof We define the function119867 on U by
119867(119911) = 119911
minus119901D(120582119901)
119905 (119899119898)F (119911) (74)
Differentiation of both the sides of (74) combined with theidentity
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
119901
= D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)(75)
gives
119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
) = 119867 (119911) +
119911119867
1015840(119911)
119901
(76)
By making use of (67) we simplify the subordinate of (72) asfollows
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus 120574)119867 (119911) + 120574119911
minus119901D(120582119901)
119905 (119899119898)
times (
120583
120583 + 119901
F (119911) +
119901
120583 + 119901
119911F1015840 (119911)
119901
)
= (1 minus 120574)119867 (119911)
120583120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+
119901120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (
119911F1015840 (119911)
119901
)
(77)
Next by using (74) and (76) the above identity furthersimplifies to the following
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus
119901120574
120583 + 119901
)119867 (119911) +
119901120574
120583 + 119901
(119867 (119911) +
119911119867
1015840(119911)
119901
)
= 119867 (119911) +
120574
120583 + 119901
119911119867
1015840(119911)
(78)
The subordination (72) is thus equivalent to
119867(119911) +
120574
120583 + 119901
119911119867
1015840(119911) ≺ ℎ (119911)
(R (120583) gt minus119901 120574 gt 0 119911 isin U)
(79)
Therefore an application of Lemma 2 yields the assertionof Theorem 13 The proof of Theorem 13 is completed
5 Inclusion Theorems
Theorem 14 Let 119905 gt 0 120574 gt 0 and 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574)
If 120574 le 1205740 then 119891(119911) isin H119899119898119901(120582 0 ℎ) where
1205740 =1
2
(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
minus1
(80)
The bound 1205740 is sharp when ℎ(119911) = 1(1 minus 119911)
Proof Let 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) By setting
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) (119911 isin U) (81)
we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119892 (119911) +
119905
119901
119911119892
1015840(119911)
(82)
Therefore by using Definition 1 we get
119892 (119911) +
119905
119901
119911119892
1015840(119911) ≺ 120574ℎ (119911) + 1 minus 120574 (83)
and an application of Lemma 2 yields
119892 (119911) ≺
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1ℎ (120585) 119889120585 + 1 minus 120574 = (ℎ lowast 120595) (119911)
(84)
where
120595 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574 (85)
It follows from (85) that if 0 lt 120574 le 1205740 where 1205740 gt 1 is givenby (80) then
R (120595 (119911)) =
120574119901
119905
int
1
0
119906
(119901119905)minus1R(
1
1 minus 119906119911
) 119889119906 + 1 minus 120574
gt
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906 + 1 minus 120574
ge
1
2
(119911 isin U)
(86)
Equivalently
120595 (119911) ≺
1
1 minus 119911
(119911 isin U) (87)
Since ℎ(119911) and 120595(119911) are both convex univalent functions inU using Lemma 4 we obtain from (84) that
119892 (119911) ≺ (ℎ lowast 120595) (119911) ≺ ℎ (119911) (88)
Therefore in view of (81) we have
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (89)
Or equivalently 119891(119911) isin H119899119898119901(120582 0 ℎ)
In order to prove that the bound on 1205740 is the best possibleset ℎ(119911) = 1(1 minus 119911) and let the function 119891 be defined onU by
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574
(90)
8 Journal of Complex Analysis
It can be readily verified that 119891 isin A119901 and
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 120574ℎ (119911) + 1 minus 120574
(91)
Thus 119891(119911) isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) Also for 120574 gt 1205740 and
119911 rarr minus1 we have
R 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) 997888rarr
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906
+ 1 minus 120574 lt
1
2
(92)
which shows that
119891 (119911) notin H119899119898
119901(120582 0 ℎ) (93)
Hence the bound 1205740 cannot be increased when ℎ(119911) = 1(1 minus119911) The proof of Theorem 14 is completed
Taking119898 = 0 we use the notation
H1198990
119901(120582 119905 ℎ) = H
119899
119901(120582 119905 ℎ) (94)
in the following theorem
Theorem 15 Let 0 le 1199051 lt 1199052 Then
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (95)
Proof Let 0 le 1199051 lt 1199052 and suppose that 119891 isin H119899119901(120582 1199052 ℎ) We
define
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899 0) 119891 (119911) (119911 isin U) (96)
The function 119892 is analytic in U with 119892(0) = 1 Differenti-ating the expressions on both the sides of (96) with respect to119911 and using Definition 1 we have
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
= 119892 (119911) +
1199052
119901
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U)
(97)
Hence an application of Lemma 2 yields
119892 (119911) ≺ ℎ (119911) (119911 isin U) (98)
Noting that 0 le 11990511199052 lt 1 and ℎ(119911) is convex univalent in Uit follows from (96) (97) and (98) that
(1 minus 1199051) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199051
119911
minus119901+1(D(120582119901)
119905 (119899 0)119891(119911))
1015840
=
1199051
1199052
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
+ (1 minus
1199051
1199052
)119892 (119911)
≺ ℎ (119911) (119911 isin U)
(99)
Thus 119891 isin H119899119901(120582 1199051 ℎ) Consequently
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (100)
The proof of Theorem 15 is completed
Remark 16 The particular case 119899 = 1 of Theorem 15 gives aresult of Liu ([17] Theorem 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Patel and A K Mishra ldquoOn certain subclasses of multivalentfunctions associated with an extended fractional differintegraloperatorrdquo Journal of Mathematical Analysis and Applicationsvol 332 no 1 pp 109ndash122 2007
[2] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987
[3] H M Srivastava and S Owa ldquoAn application of the fractionalderivativerdquo Mathematica Japonica vol 29 no 3 pp 383ndash3891984
[4] A K Mishra and P Gochhayat ldquoApplications of the Owa-Srivastava operator to the class of 119896-uniformly convex func-tionsrdquo Fractional Calculus amp Applied Analysis vol 9 no 4 pp323ndash331 2006
[5] A K Mishra and P Gochhayat ldquoThe Fekete-Szego problem fork-uniformly convex functions and for a class defined by theOwa-Srivastava operatorrdquo Journal ofMathematical Analysis andApplications vol 347 no 2 pp 563ndash572 2008
[6] H M Srivastava A K Mishra and S N Kund ldquoCertainclasses of analytic functions associated with iterations of theOwa-Srivastava fractional derivative operatorrdquo Southeast AsianBulletin of Mathematics vol 37 no 3 pp 413ndash435 2013
[7] HM Srivastava andMK Aouf ldquoA certain fractional derivativeoperator and its applications to a new class of analytic andmultivalent functions with negative coefficientsrdquo Journal ofMathematical Analysis and Applications vol 171 no 1 pp 1ndash131992
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Complex Analysis
Furthermore
119911(Ω
(120582119901)
119911119891 (119911))
1015840
= (119901 minus 120582)Ω
(120582+1119901)
119911119891 (119911)
+ 120582Ω
(120582119901)
119911119891 (119911) (minusinfin lt 120582 lt 119901 119911 isin U)
(7)
The 119899-iterates of the operatorΩ(120582119901)119911
are defined as follows
Ω
(120582119901)
1199110119891 (119911) = 119891 (119911) (8)
and for 119899 isin N
Ω
(120582119901)
119911119899119891 (119911) = Ω
(120582119901)
119911(Ω
(120582119901)
119911119899minus1119891 (119911))
= 119911
119901+
infin
sum
119896=1
[
Γ(119896 + 119901 + 1)Γ(119901 + 1 minus 120582)
Γ(119901 + 1)Γ(119896 + 119901 + 1 minus 120582)
]
119899
119886119896119911119896+119901
(9)
Similarly for 119891 isin A119901 represented by (2) let the operator
C119898
119905 A119901 997888rarr A119901 (119898 isin N 119905 isin C) (10)
be defined by the following
C0
119905119891 (119911) = 119891 (119911)
C1
119905119891 (119911) = (1 minus 119905) 119891 (119911) +
119905
119901
(119911119891
1015840(119911))
= 119911
119901+
infin
sum
119896=1
1 +
119905119896
119901
119886119896119911119896+119901
(11)
and for119898 isin N 1
C119898
119905119891 (119911) = C
1
119905(C119898minus1
119905119891 (119911)) = 119911
119901+
infin
sum
119896=1
1 +
119905119896
119901
119898
119886119896119911119896+119901
(12)
Very recently Srivastava et al [6] considered the compositionof the operatorsC119898
119905and Ω(120582119901)
119911119899and introduced the following
operator
D(120582119901)
119905 (119899119898) A119901 997888rarr A119901 (119899119898 isin N 1 119905 isin C) (13)
That is for 119891 isin A119901 given by (2) we know that
D(120582119901)
119905 (119899119898) 119891 (119911)
= Ω
(120582119901)
119911119899(C119898
119905119891 (119911))
= C119898
119905(Ω
(120582119901)
119911119899119891 (119911))
= 119911
119901+
infin
sum
119896=1
[
Γ (119896 + 119901 + 1) Γ (119901 + 1 minus 120582)
Γ (119901 + 1) Γ (119896 + 119901 + 1 minus 120582)
]
119899
times 1 +
119905119896
119901
119898
119886119896119911119896+119901
(119899119898 isin N0 = N cup 0 minusinfin lt 120582 lt 119901 + 1 119905 ≧ 0 119911 isin U)
(14)
The transformation D(120582119901)
119905 (119899119898) includes among many thefollowing two previously studied interesting operators asparticular cases
(i) For 119899 = 119898 isin N 119901 = 1 0 le 120582 lt 1 the frac-tional derivative operator D(1205821)
119905(119899 119899) = D119899120582
119905was
recently introduced and investigated by Al-Oboudiand Al-Amoudi [9 10] in the context of functionsrepresented by conical domains
(ii) For 119899 = 119898 isin N 119901 = 1 120582 = 0 119905 = 1 D(01)1
(119899 119899) is the Salagean operator [11] which is in factthe 119899-iterates of the popular Alexanderrsquos differentialtransform 119891(119911) rarr 119911119891
1015840(119911) [12]
We next recall the definition of subordination Supposethat 119891 isin A and 119892 inA is univalent in U We say that 119891(119911) issubordinate to 119892(119911) in U if 119891(0) = 119892(0) and 119891(U) sube 119892(U)Considering the function 119908(119911) = 119892
minus1(119891(119911)) it is readily
checked that 119908(119911) satisfies the conditions of the Schwarzlemma and
119891 (119911) = 119892 (119908 (119911)) (119911 isin U) (15)
In a broader sense the function 119891 isin A is said to be sub-ordinate to the function 119892 isin A (119892 need not be univalent inU) written as
119891 ≺ 119892 in U or 119891 (119911) ≺ 119892 (119911) (119911 isin U) (16)
if condition (15) holds for some Schwarz function 119908(119911) (see[12] for details) We also need the following definition ofHadamard product (or convolution) For the functions 119891 and119892 inA119901 given by the following Taylor-Maclaurinrsquos series
119891 (119911) = 119911
119901+
infin
sum
119896=1
119886119896119911119896+119901
119892 (119911) = 119911
119901+
infin
sum
119896=1
119887119896119911119896+119901
(119911 isin U)
(17)
their Hadamard product (or convolution) 119891 lowast 119892 is defined by
(119891 lowast 119892) (119911) = 119911
119901+
infin
sum
119896=1
119886119896119887119896119911119896+119901
(119911 isin U) (18)
It is easy to see that 119891 lowast 119892 isin A119901The study of iterations of entire and meromorphic func-
tions as the number of iterations tends to infinity is a populartopic in complex analysis However investigations havebeen initiated only recently regarding iterations of certaintransforms defined on classes of analytic and meromorphicfunctions For example Al-Oboudi and Al-Amoudi [9 10]investigated properties of certain classes of analytic functionsassociated with conical domains by making use of theoperator D119899120582
119905 Their work generalized several earlier results
of Srivastava and Mishra [13] This theme has been furtherpursued in our more recent papers [6 14 15] In the sequel to
Journal of Complex Analysis 3
these current investigations in the present paper we definethe following subclass of A119901 associated with the iteratedoperator D(120582119901)119905 (119899119898) and investigate its several interestingproperties Our work is also motivated by earlier works in[16ndash20] connecting subordination and Hadamard product
Definition 1 The function 119891 isin A119901 is said to be in theclassH119899119898
119901(120582 119905 ℎ) if the following subordination condition is
satisfied
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
≺ ℎ (119911)
(119911 isin U)
(19)
where 119905 is a complex number and ℎ is an analytic convexunivalent function in U
The function class H119899119898119901(120582 119905 ℎ) includes several previ-
ously studied subclasses ofA as particular cases For example
(i) for 119899 = 1 119898 = 0 the classH10119901(120582 119905 ℎ) = H119901(120582 119905 ℎ)
was recently studied by Liu [17](ii) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 119905 = 1 and ℎ(119911) =
(1 + 119911)(1 minus 119911) the classH101(1 1 (1 + 119911)(1 minus 119911)) was
earlier investigated by R Singh and S Singh [21](iii) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 and ℎ(119911) = (1 +
119886119911)(1+119887119911) (minus1 le 119887 lt 119886 le 1) H101(1 119905 (1+119886119911)(1+
119887119911)) reduces to H(119905 119886 119887) which was investigated byYang [22]
(iv) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 and ℎ(119911) =
1+119872119911 (119872 gt 0) H101(1 119905 1+119872119911) reduces toS(119905119872)
the class introduced and studied by Zhongzhu andOwa [23] and Jinlin [24]
In the present paper we primarily focus on a variety ofconvolution theorems for the class H119899119898
119901(120582 119905 ℎ) We also
find inclusion theorems and study behavior of the Libera-Livingston integral operator
2 Some More Definitions andPreliminary Lemmas
We need the following definitions and results for the presen-tation of our results Let CV(120588) and Slowast(120588) (0 le 120588 lt 1)
denote respectively the classes of univalent convex functionsof order 120588 and starlike functions of order 120588 (see [12] for details)The function 119891 isin A1 is said to be in the class PSlowast(120588)consisting of prestarlike functions of order 120588 [25] if
119911
(1 minus 119911)
2(1minus120588)lowast 119891 (119911) isin S
lowast(120588) (20)
It is readily seen that
PSlowast(0) = CV (0) = CV PS
lowast(
1
2
) = Slowast(
1
2
)
(21)
Furthermore it is well known [25] that
PSlowast(120583) sub PS
lowast(120582) (120583 le 120582) (22)
Wewill also need the following lemmas in order to derive ourmain results
Lemma 2 (see [26] also see [27]) Let 119892 be analytic in U andlet ℎ be analytic and convex univalent in U with ℎ(0) = 119892(0)If
119892 (119911) +
1
120583
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U) (23)
whereR(120583) ge 0 and 120583 = 0 then
119892 (119911) ≺ℎ (119911) = 120583119911
minus120583int
119911
0
120585
120583minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(24)
and ℎ is the best dominant of (24)
Lemma 3 (see [25]) Let 120588 lt 1 119891 isin Slowast(120588) and 119892 isin PSlowast(120588)Then for any analytic functionF in U
119892 lowast (119891F)
119892 lowast 119891
(U) sub 119862119874 (F (U)) (25)
where 119862119874(F(U)) denotes the closed convex hull ofF(U)
Lemma 4 (see [28]) Let 119867 and 119866 be univalent convexfunctions inU and let ℎ and 119892 be functions inA Suppose thatℎ ≺ 119867 and 119892 ≺ 119866 in U Then ℎ lowast 119892 ≺ 119867 lowast 119866 in U
The following well known result is a consequence of theprinciple of subordination and can be found for example in[12 29]
Lemma 5 Let the function 119892 isin A satisfy 119892(0) = 1 andR(119892(119911)) gt 1205730 (0 le 1205730 lt 1 119911 isin U) Then
R (119892 (119911)) gt 1205730 + (1 minus 1205730)1 minus |119911|
1 + |119911|
(119911 isin U) (26)
3 Convolution Results
We state and prove the following convolution results
Theorem 6 Let 119891 isin H119899119898119901(120582 119905 ℎ) and suppose that 119892 in A119901
satisfies the following
R 119911
minus119901119892 (119911) gt
1
2
(119911 isin U) (27)
Then
119891 lowast 119892 isin H119899119898
119901(120582 119905 ℎ) (28)
4 Journal of Complex Analysis
Proof For every 119891 and 119892 inA119901 we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
= (1 minus 119905) (119911
minus119901119892 (119911)) lowast (119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911))
+
119905
119901
(119911
minus119901119892 (119911)) lowast (119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
)
= (119911
minus119901119892 (119911)) lowast Ψ (119911)
(29)
where
Ψ (119911) = (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
(30)
Now if 119891 isin H119899119898119901(120582 119905 ℎ) then
Ψ (119911) ≺ ℎ (119911) (119911 isin U) (31)
Furthermore condition (27) is equivalent to
119911
minus119901119892 (119911) ≺
1
1 minus 119911
(119911 isin U) (32)
Therefore an application of Lemma 4 in (29) yields
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
≺
1
1 minus 119911
lowast ℎ (119911)
= ℎ (119911)
(33)
This shows that 119891 lowast 119892 isin H119899119898119901(120582 119905 ℎ) The proof of
Theorem 6 is completed
Remark 7 Taking 119899 = 1 119898 = 0 inTheorem 6 we get a recentresult of Liu ([17]Theorem 3)The choices 119899 = 1 119898 = 0 119901 =1 120582 = 1 and ℎ(119911) = (1 + 119886119911)(1 + 119887119911) (minus1 le 119887 lt 1 119886 gt 119887)
yield a result of Yang ([22] Theorem 4)
Corollary 8 Let the function 119891 given by (2) be a member ofH119899119898119901(120582 119905 ℎ) and
119904119903 (119911) = 119911119901+
119903minus1
sum
119896=1
119886119896119911119896+119901
(119903 isin N 1 119911 isin U) (34)
Then the function
120590119903 (119911) = int
1
0
119905
minus119901119904119903 (119905119911) 119889119905
(35)
is in the classH119899119898119901(120582 119905 ℎ)
Proof Let 119891 isin H119899119898119901(120582 119905 ℎ) We note that
120590119903 (119911) = 119911119901+
119903minus1
sum
119896=1
119886119896
119896 + 1
119911
119896+119901= (119891 lowast 119892119903) (119911) (119903 isin N 1)
(36)
where
119891 (119911) = 119911
119901+
infin
sum
119896=1
119886119896119911119896+119901
119892119903 (119911) = 119911
119901+
119903minus1
sum
119896=1
119911
119896+119901
119896 + 1
isin A119901
(37)
Also for 119903 isin N 1 it is well known [18] that
R 119911
minus119901119892119903 (119911) gt
1
2
(119911 isin U) (38)
In view of (36) and (38) an application of Theorem 6 gives
120590119903 isin H119899119898
119901(120582 119905 ℎ) (39)
The proof of Corollary 8 is completed
Theorem 9 Let the function 119892 in A119901 be such that 119911minus119901+1119892(119911)is a prestarlike function of order 120588 (120588 lt 1) If119891 isin H119899119898
119901(120582 119905 ℎ)
then119891 lowast 119892 isin H
119899119898
119901(120582 119905 ℎ) (40)
Proof Let 119891 isin H119899119898119901(120582 119905 ℎ) and 119892 isin A119901 Then (29) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
=
(119911
minus119901+1119892 (119911)) lowast (119911Ψ (119911))
(119911
minus119901+1119892 (119911)) lowast 119911
(119911 isin U)
(41)
where Ψ(119911) is defined as in (30) We noted in the proof ofTheorem 6 thatΨ(119911) ≺ ℎ(119911) Since 119911minus119901+1119892(119911) isin PSlowast(120588) 119911 isinSlowast(120588) and ℎ(119911) is convex univalent in U an application ofLemma 3 in (41) yields the following
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)(119891 lowast 119892)(119911))
1015840
≺ ℎ (119911) (119911 isin U)
(42)
Therefore 119891 lowast 119892 isin H119899119898119901(120582 119905 ℎ) The proof Theorem 9 is
completed
Taking 120588 = 12 in Theorem 9 we get the following
Corollary 10 Let 119891 isin H119899119898119901(120582 119905 ℎ) and suppose that 119892 inA119901
is such that 119911minus119901+1119892 isin Slowast(12) Then
119891 lowast 119892 isin H119899119898
119901(120582 119905 ℎ) (43)
In particular if 119911minus119901+1119892 is univalent convex then 119891 lowast 119892 isin
H119899119898119901(120582 119905 ℎ)
Journal of Complex Analysis 5
In the following theorem we discuss convolution proper-ties of the function classH(120582 119905 ℎ)when ℎ is a right half planemapping
Theorem 11 Let 119905 ge 0 and suppose that each of the functions119891119895 (119895 = 1 2) is a member of the classH119899119898
119901(120582 119905 ℎ119895) where
ℎ119895 (119911) =
1 + (1 minus 2120573119895) 119911
1 minus 119911
(0 le 120573119895 lt 1) (44)
If 119891 isin A119901 is defined by the following
119891 (119911) = D(120582119901)
119905 (119899119898) (1198911 lowast 1198912) (119911) (119911 isin U) (45)
then 119891 isin H119899119898119901(120582 119905 ℎ) where
ℎ (119911) =
1 + (1 minus 2120573) 119911
1 minus 119911
(46)
and 120573 is given by
120573 =
1 minus 4 (1 minus 1205731) (1 minus 1205732)
times(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906) (120582 gt 0)
1 minus 2 (1 minus 1205731) (1 minus 1205732) (120582 = 0)
(47)
The bound on 120573 is the best possible
Proof We consider the case 119905 gt 0 Suppose that 119891119895 isin
H119899119898119901(120582 119905 ℎ119895) (119895 = 1 2) where ℎ119895(119911) is given by (44) By
setting
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891119895(119911))
1015840(48)
we see in the light of Definition 1 that
Φ119895 (119911) ≺ ℎ119895 (119911) (119895 = 1 2) (49)
A routine calculation yields the following
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1Φ119895 (120585) 119889120585
(119895 = 1 2)
(50)
Now if 119891(119911) is defined by (45) then using (50) we get that
D(120582119901)
119905 (119899119898) 119891 (119911)
= D(120582119901)
119905 (119899119898) 1198911 (119911) lowastD(120582119901)
119905 (119899119898) 1198912 (119911)
= (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ1 (119906119911) 119889119906)
lowast (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ2 (119906119911) 119889119906)
=
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ0 (119906119911) 119889119906
(51)
where
Φ0 (119911) =
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1(Φ1 lowast Φ2) (119906119911) 119889119906
(52)
Since
Φ1 (119911) minus 1205731
1 minus 1205731
≺
1 + 119911
1 minus 119911
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
≺
1
1 minus 119911
(53)
by using Lemma 4 we get
(
Φ1 (119911) minus 1205731
1 minus 1205731
) lowast (
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
) ≺
1 + 119911
1 minus 119911
(54)
A simple calculation gives that
Φ1 (119911) lowast Φ2 (119911) ≺ 1 plusmn 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(55)
Therefore by the Lindeloff principle of subordinationwehave
R Φ1 (119911) lowast Φ2 (119911)
gt 1205730 = 1 minus 2 (1 minus 1205731) (1 minus 1205732) (119911 isin U) (56)
Since
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= Φ0 (119911)
(57)
by using (52) in conjunction with (56) and Lemma 4 we getthe following
R (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= R Φ0 (119911)
=
119901
119905
int
1
0
119906
(119901119905)minus1R (Φ1 lowast Φ2) (119906119911) 119889119906
ge
119901
119905
int
1
0
119906
(119901119905)minus1(1205730 + (1 minus 1205730)
1 minus 119906 |119911|
1 + 119906 |119911|
) 119889119906
gt 1205730 +
119901 (1 minus 1205730)
119905
int
1
0
119906
(119901119905)minus1 1 minus 119906
1 + 119906
119889119906
= 1 minus 4 (1 minus 1205731) (1 minus 1205732) (1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
= 120573 (119911 isin U)
(58)
This proves that119891(119911) isin H119899119898119901(120582 119905 ℎ) where the function ℎ(119911)
is given by (46)
6 Journal of Complex Analysis
In order to show that the value of 120573 is the least possiblewe take the functions 119891119895(119911) isin A119901 (119895 = 1 2) defined by
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1
times (120573119895 + (1 minus 120573119895)1 + 120585
1 minus 120585
) 119889120585
(59)
for which we have
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891119895 (119911))
1015840
= 120573119895 + (1 minus 120573119895)1 + 119911
1 minus 119911
(119895 = 1 2)
(60)
(Φ1 lowast Φ2) (119911) = 1 + 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(61)
Hence for 119891 isin A119901 given by (45) we obtain
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
=
119901
119905
int
1
0
119906
(119901119905)minus1(1 + 4 (1 minus 1205731) (1 minus 1205732)
119906119911
1 minus 119906119911
) 119889119906
997888rarr 120573 (as 119911 997888rarr minus1)
(62)
Finally for the case 119905 = 0 the proof of Theorem 11 is simpleso we choose to omit the details involved
4 Properties of the Libera-LivingstonTransform
For the function 119891 isin A119901 the functionF defined by
F (119911) =
120583 + 119901
119911
120583int
119911
0
120585
120583minus1119891 (120585) 119889120585 (R (120583) gt minus119901 119911 isin U)
(63)
is popularly known as the Libera-Livingston transform of 119891We state and prove the following
Theorem 12 Let 119891 isin H119899119898119901(120582 119905 ℎ) Then the function F
defined by (63) is in the classH119899119898119901(120582 119905
ℎ) where
ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
120585
120583+119901minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(64)
ConsequentlyF isin H119899119898119901(120582 119905 ℎ) The function ℎ(119911) is the best
dominant in (64)
Proof We define the function119870 on U by
119911
119901119870 (119911) = (1 minus 119905)D
(120582119901)
119905 (119899119898)F (119911)
+
119905
119901
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
(65)
Differentiating both the sides of (65) with respect to 119911 we get
119901119870 (119911) + 119911119870
1015840(119911)
= 119901 (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)
+ 119905119911
minus119901+1(D(120582119901)
119905 (119899119898)
119911F1015840(119911)
119901
)
1015840
(66)
Also the defining relation (63) yields
(120583 + 119901) 119891 (119911) = 120583F (119911) + 119911F1015840(119911) (67)
Now a routine calculation using (65) (66) and (67) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
(119911 isin U)
(68)
Since 119891 isin H119899119898119901(120582 119905 ℎ) we get the following from the
preceding equation (68)
119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
≺ ℎ (119911) (69)
Therefore by applying Lemma 2 we have
119870 (119911) ≺ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
119905
120583+119901minus1ℎ (119905) 119889119905 ≺ ℎ (119911)
(70)
This last subordination (70) is equivalent to
F isin H119899119898
119901(120582 119905
ℎ) sub H
119899119898
119901(120582 119905 ℎ) (71)
The proof of Theorem 12 is completed
Theorem 13 Let 119891 isin A119901 and suppose that the function F isdefined as in (63) If
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (120574 gt 0)
(72)
thenF isin H119899119898119901(120582 0
ℎ) where
ℎ (119911) =
120583 + 119901
120574
119911
minus(120583+119901)120574
times int
119911
0
120585
((120583+119901)120574)minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (R (120583) gt minus119901)
(73)
The function ℎ(119911) is the best dominant in (73)
Journal of Complex Analysis 7
Proof We define the function119867 on U by
119867(119911) = 119911
minus119901D(120582119901)
119905 (119899119898)F (119911) (74)
Differentiation of both the sides of (74) combined with theidentity
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
119901
= D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)(75)
gives
119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
) = 119867 (119911) +
119911119867
1015840(119911)
119901
(76)
By making use of (67) we simplify the subordinate of (72) asfollows
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus 120574)119867 (119911) + 120574119911
minus119901D(120582119901)
119905 (119899119898)
times (
120583
120583 + 119901
F (119911) +
119901
120583 + 119901
119911F1015840 (119911)
119901
)
= (1 minus 120574)119867 (119911)
120583120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+
119901120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (
119911F1015840 (119911)
119901
)
(77)
Next by using (74) and (76) the above identity furthersimplifies to the following
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus
119901120574
120583 + 119901
)119867 (119911) +
119901120574
120583 + 119901
(119867 (119911) +
119911119867
1015840(119911)
119901
)
= 119867 (119911) +
120574
120583 + 119901
119911119867
1015840(119911)
(78)
The subordination (72) is thus equivalent to
119867(119911) +
120574
120583 + 119901
119911119867
1015840(119911) ≺ ℎ (119911)
(R (120583) gt minus119901 120574 gt 0 119911 isin U)
(79)
Therefore an application of Lemma 2 yields the assertionof Theorem 13 The proof of Theorem 13 is completed
5 Inclusion Theorems
Theorem 14 Let 119905 gt 0 120574 gt 0 and 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574)
If 120574 le 1205740 then 119891(119911) isin H119899119898119901(120582 0 ℎ) where
1205740 =1
2
(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
minus1
(80)
The bound 1205740 is sharp when ℎ(119911) = 1(1 minus 119911)
Proof Let 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) By setting
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) (119911 isin U) (81)
we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119892 (119911) +
119905
119901
119911119892
1015840(119911)
(82)
Therefore by using Definition 1 we get
119892 (119911) +
119905
119901
119911119892
1015840(119911) ≺ 120574ℎ (119911) + 1 minus 120574 (83)
and an application of Lemma 2 yields
119892 (119911) ≺
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1ℎ (120585) 119889120585 + 1 minus 120574 = (ℎ lowast 120595) (119911)
(84)
where
120595 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574 (85)
It follows from (85) that if 0 lt 120574 le 1205740 where 1205740 gt 1 is givenby (80) then
R (120595 (119911)) =
120574119901
119905
int
1
0
119906
(119901119905)minus1R(
1
1 minus 119906119911
) 119889119906 + 1 minus 120574
gt
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906 + 1 minus 120574
ge
1
2
(119911 isin U)
(86)
Equivalently
120595 (119911) ≺
1
1 minus 119911
(119911 isin U) (87)
Since ℎ(119911) and 120595(119911) are both convex univalent functions inU using Lemma 4 we obtain from (84) that
119892 (119911) ≺ (ℎ lowast 120595) (119911) ≺ ℎ (119911) (88)
Therefore in view of (81) we have
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (89)
Or equivalently 119891(119911) isin H119899119898119901(120582 0 ℎ)
In order to prove that the bound on 1205740 is the best possibleset ℎ(119911) = 1(1 minus 119911) and let the function 119891 be defined onU by
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574
(90)
8 Journal of Complex Analysis
It can be readily verified that 119891 isin A119901 and
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 120574ℎ (119911) + 1 minus 120574
(91)
Thus 119891(119911) isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) Also for 120574 gt 1205740 and
119911 rarr minus1 we have
R 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) 997888rarr
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906
+ 1 minus 120574 lt
1
2
(92)
which shows that
119891 (119911) notin H119899119898
119901(120582 0 ℎ) (93)
Hence the bound 1205740 cannot be increased when ℎ(119911) = 1(1 minus119911) The proof of Theorem 14 is completed
Taking119898 = 0 we use the notation
H1198990
119901(120582 119905 ℎ) = H
119899
119901(120582 119905 ℎ) (94)
in the following theorem
Theorem 15 Let 0 le 1199051 lt 1199052 Then
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (95)
Proof Let 0 le 1199051 lt 1199052 and suppose that 119891 isin H119899119901(120582 1199052 ℎ) We
define
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899 0) 119891 (119911) (119911 isin U) (96)
The function 119892 is analytic in U with 119892(0) = 1 Differenti-ating the expressions on both the sides of (96) with respect to119911 and using Definition 1 we have
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
= 119892 (119911) +
1199052
119901
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U)
(97)
Hence an application of Lemma 2 yields
119892 (119911) ≺ ℎ (119911) (119911 isin U) (98)
Noting that 0 le 11990511199052 lt 1 and ℎ(119911) is convex univalent in Uit follows from (96) (97) and (98) that
(1 minus 1199051) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199051
119911
minus119901+1(D(120582119901)
119905 (119899 0)119891(119911))
1015840
=
1199051
1199052
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
+ (1 minus
1199051
1199052
)119892 (119911)
≺ ℎ (119911) (119911 isin U)
(99)
Thus 119891 isin H119899119901(120582 1199051 ℎ) Consequently
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (100)
The proof of Theorem 15 is completed
Remark 16 The particular case 119899 = 1 of Theorem 15 gives aresult of Liu ([17] Theorem 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Patel and A K Mishra ldquoOn certain subclasses of multivalentfunctions associated with an extended fractional differintegraloperatorrdquo Journal of Mathematical Analysis and Applicationsvol 332 no 1 pp 109ndash122 2007
[2] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987
[3] H M Srivastava and S Owa ldquoAn application of the fractionalderivativerdquo Mathematica Japonica vol 29 no 3 pp 383ndash3891984
[4] A K Mishra and P Gochhayat ldquoApplications of the Owa-Srivastava operator to the class of 119896-uniformly convex func-tionsrdquo Fractional Calculus amp Applied Analysis vol 9 no 4 pp323ndash331 2006
[5] A K Mishra and P Gochhayat ldquoThe Fekete-Szego problem fork-uniformly convex functions and for a class defined by theOwa-Srivastava operatorrdquo Journal ofMathematical Analysis andApplications vol 347 no 2 pp 563ndash572 2008
[6] H M Srivastava A K Mishra and S N Kund ldquoCertainclasses of analytic functions associated with iterations of theOwa-Srivastava fractional derivative operatorrdquo Southeast AsianBulletin of Mathematics vol 37 no 3 pp 413ndash435 2013
[7] HM Srivastava andMK Aouf ldquoA certain fractional derivativeoperator and its applications to a new class of analytic andmultivalent functions with negative coefficientsrdquo Journal ofMathematical Analysis and Applications vol 171 no 1 pp 1ndash131992
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
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Journal of Complex Analysis 3
these current investigations in the present paper we definethe following subclass of A119901 associated with the iteratedoperator D(120582119901)119905 (119899119898) and investigate its several interestingproperties Our work is also motivated by earlier works in[16ndash20] connecting subordination and Hadamard product
Definition 1 The function 119891 isin A119901 is said to be in theclassH119899119898
119901(120582 119905 ℎ) if the following subordination condition is
satisfied
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
≺ ℎ (119911)
(119911 isin U)
(19)
where 119905 is a complex number and ℎ is an analytic convexunivalent function in U
The function class H119899119898119901(120582 119905 ℎ) includes several previ-
ously studied subclasses ofA as particular cases For example
(i) for 119899 = 1 119898 = 0 the classH10119901(120582 119905 ℎ) = H119901(120582 119905 ℎ)
was recently studied by Liu [17](ii) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 119905 = 1 and ℎ(119911) =
(1 + 119911)(1 minus 119911) the classH101(1 1 (1 + 119911)(1 minus 119911)) was
earlier investigated by R Singh and S Singh [21](iii) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 and ℎ(119911) = (1 +
119886119911)(1+119887119911) (minus1 le 119887 lt 119886 le 1) H101(1 119905 (1+119886119911)(1+
119887119911)) reduces to H(119905 119886 119887) which was investigated byYang [22]
(iv) for 119899 = 1 119898 = 0 119901 = 1 120582 = 1 and ℎ(119911) =
1+119872119911 (119872 gt 0) H101(1 119905 1+119872119911) reduces toS(119905119872)
the class introduced and studied by Zhongzhu andOwa [23] and Jinlin [24]
In the present paper we primarily focus on a variety ofconvolution theorems for the class H119899119898
119901(120582 119905 ℎ) We also
find inclusion theorems and study behavior of the Libera-Livingston integral operator
2 Some More Definitions andPreliminary Lemmas
We need the following definitions and results for the presen-tation of our results Let CV(120588) and Slowast(120588) (0 le 120588 lt 1)
denote respectively the classes of univalent convex functionsof order 120588 and starlike functions of order 120588 (see [12] for details)The function 119891 isin A1 is said to be in the class PSlowast(120588)consisting of prestarlike functions of order 120588 [25] if
119911
(1 minus 119911)
2(1minus120588)lowast 119891 (119911) isin S
lowast(120588) (20)
It is readily seen that
PSlowast(0) = CV (0) = CV PS
lowast(
1
2
) = Slowast(
1
2
)
(21)
Furthermore it is well known [25] that
PSlowast(120583) sub PS
lowast(120582) (120583 le 120582) (22)
Wewill also need the following lemmas in order to derive ourmain results
Lemma 2 (see [26] also see [27]) Let 119892 be analytic in U andlet ℎ be analytic and convex univalent in U with ℎ(0) = 119892(0)If
119892 (119911) +
1
120583
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U) (23)
whereR(120583) ge 0 and 120583 = 0 then
119892 (119911) ≺ℎ (119911) = 120583119911
minus120583int
119911
0
120585
120583minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(24)
and ℎ is the best dominant of (24)
Lemma 3 (see [25]) Let 120588 lt 1 119891 isin Slowast(120588) and 119892 isin PSlowast(120588)Then for any analytic functionF in U
119892 lowast (119891F)
119892 lowast 119891
(U) sub 119862119874 (F (U)) (25)
where 119862119874(F(U)) denotes the closed convex hull ofF(U)
Lemma 4 (see [28]) Let 119867 and 119866 be univalent convexfunctions inU and let ℎ and 119892 be functions inA Suppose thatℎ ≺ 119867 and 119892 ≺ 119866 in U Then ℎ lowast 119892 ≺ 119867 lowast 119866 in U
The following well known result is a consequence of theprinciple of subordination and can be found for example in[12 29]
Lemma 5 Let the function 119892 isin A satisfy 119892(0) = 1 andR(119892(119911)) gt 1205730 (0 le 1205730 lt 1 119911 isin U) Then
R (119892 (119911)) gt 1205730 + (1 minus 1205730)1 minus |119911|
1 + |119911|
(119911 isin U) (26)
3 Convolution Results
We state and prove the following convolution results
Theorem 6 Let 119891 isin H119899119898119901(120582 119905 ℎ) and suppose that 119892 in A119901
satisfies the following
R 119911
minus119901119892 (119911) gt
1
2
(119911 isin U) (27)
Then
119891 lowast 119892 isin H119899119898
119901(120582 119905 ℎ) (28)
4 Journal of Complex Analysis
Proof For every 119891 and 119892 inA119901 we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
= (1 minus 119905) (119911
minus119901119892 (119911)) lowast (119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911))
+
119905
119901
(119911
minus119901119892 (119911)) lowast (119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
)
= (119911
minus119901119892 (119911)) lowast Ψ (119911)
(29)
where
Ψ (119911) = (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
(30)
Now if 119891 isin H119899119898119901(120582 119905 ℎ) then
Ψ (119911) ≺ ℎ (119911) (119911 isin U) (31)
Furthermore condition (27) is equivalent to
119911
minus119901119892 (119911) ≺
1
1 minus 119911
(119911 isin U) (32)
Therefore an application of Lemma 4 in (29) yields
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
≺
1
1 minus 119911
lowast ℎ (119911)
= ℎ (119911)
(33)
This shows that 119891 lowast 119892 isin H119899119898119901(120582 119905 ℎ) The proof of
Theorem 6 is completed
Remark 7 Taking 119899 = 1 119898 = 0 inTheorem 6 we get a recentresult of Liu ([17]Theorem 3)The choices 119899 = 1 119898 = 0 119901 =1 120582 = 1 and ℎ(119911) = (1 + 119886119911)(1 + 119887119911) (minus1 le 119887 lt 1 119886 gt 119887)
yield a result of Yang ([22] Theorem 4)
Corollary 8 Let the function 119891 given by (2) be a member ofH119899119898119901(120582 119905 ℎ) and
119904119903 (119911) = 119911119901+
119903minus1
sum
119896=1
119886119896119911119896+119901
(119903 isin N 1 119911 isin U) (34)
Then the function
120590119903 (119911) = int
1
0
119905
minus119901119904119903 (119905119911) 119889119905
(35)
is in the classH119899119898119901(120582 119905 ℎ)
Proof Let 119891 isin H119899119898119901(120582 119905 ℎ) We note that
120590119903 (119911) = 119911119901+
119903minus1
sum
119896=1
119886119896
119896 + 1
119911
119896+119901= (119891 lowast 119892119903) (119911) (119903 isin N 1)
(36)
where
119891 (119911) = 119911
119901+
infin
sum
119896=1
119886119896119911119896+119901
119892119903 (119911) = 119911
119901+
119903minus1
sum
119896=1
119911
119896+119901
119896 + 1
isin A119901
(37)
Also for 119903 isin N 1 it is well known [18] that
R 119911
minus119901119892119903 (119911) gt
1
2
(119911 isin U) (38)
In view of (36) and (38) an application of Theorem 6 gives
120590119903 isin H119899119898
119901(120582 119905 ℎ) (39)
The proof of Corollary 8 is completed
Theorem 9 Let the function 119892 in A119901 be such that 119911minus119901+1119892(119911)is a prestarlike function of order 120588 (120588 lt 1) If119891 isin H119899119898
119901(120582 119905 ℎ)
then119891 lowast 119892 isin H
119899119898
119901(120582 119905 ℎ) (40)
Proof Let 119891 isin H119899119898119901(120582 119905 ℎ) and 119892 isin A119901 Then (29) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
=
(119911
minus119901+1119892 (119911)) lowast (119911Ψ (119911))
(119911
minus119901+1119892 (119911)) lowast 119911
(119911 isin U)
(41)
where Ψ(119911) is defined as in (30) We noted in the proof ofTheorem 6 thatΨ(119911) ≺ ℎ(119911) Since 119911minus119901+1119892(119911) isin PSlowast(120588) 119911 isinSlowast(120588) and ℎ(119911) is convex univalent in U an application ofLemma 3 in (41) yields the following
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)(119891 lowast 119892)(119911))
1015840
≺ ℎ (119911) (119911 isin U)
(42)
Therefore 119891 lowast 119892 isin H119899119898119901(120582 119905 ℎ) The proof Theorem 9 is
completed
Taking 120588 = 12 in Theorem 9 we get the following
Corollary 10 Let 119891 isin H119899119898119901(120582 119905 ℎ) and suppose that 119892 inA119901
is such that 119911minus119901+1119892 isin Slowast(12) Then
119891 lowast 119892 isin H119899119898
119901(120582 119905 ℎ) (43)
In particular if 119911minus119901+1119892 is univalent convex then 119891 lowast 119892 isin
H119899119898119901(120582 119905 ℎ)
Journal of Complex Analysis 5
In the following theorem we discuss convolution proper-ties of the function classH(120582 119905 ℎ)when ℎ is a right half planemapping
Theorem 11 Let 119905 ge 0 and suppose that each of the functions119891119895 (119895 = 1 2) is a member of the classH119899119898
119901(120582 119905 ℎ119895) where
ℎ119895 (119911) =
1 + (1 minus 2120573119895) 119911
1 minus 119911
(0 le 120573119895 lt 1) (44)
If 119891 isin A119901 is defined by the following
119891 (119911) = D(120582119901)
119905 (119899119898) (1198911 lowast 1198912) (119911) (119911 isin U) (45)
then 119891 isin H119899119898119901(120582 119905 ℎ) where
ℎ (119911) =
1 + (1 minus 2120573) 119911
1 minus 119911
(46)
and 120573 is given by
120573 =
1 minus 4 (1 minus 1205731) (1 minus 1205732)
times(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906) (120582 gt 0)
1 minus 2 (1 minus 1205731) (1 minus 1205732) (120582 = 0)
(47)
The bound on 120573 is the best possible
Proof We consider the case 119905 gt 0 Suppose that 119891119895 isin
H119899119898119901(120582 119905 ℎ119895) (119895 = 1 2) where ℎ119895(119911) is given by (44) By
setting
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891119895(119911))
1015840(48)
we see in the light of Definition 1 that
Φ119895 (119911) ≺ ℎ119895 (119911) (119895 = 1 2) (49)
A routine calculation yields the following
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1Φ119895 (120585) 119889120585
(119895 = 1 2)
(50)
Now if 119891(119911) is defined by (45) then using (50) we get that
D(120582119901)
119905 (119899119898) 119891 (119911)
= D(120582119901)
119905 (119899119898) 1198911 (119911) lowastD(120582119901)
119905 (119899119898) 1198912 (119911)
= (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ1 (119906119911) 119889119906)
lowast (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ2 (119906119911) 119889119906)
=
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ0 (119906119911) 119889119906
(51)
where
Φ0 (119911) =
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1(Φ1 lowast Φ2) (119906119911) 119889119906
(52)
Since
Φ1 (119911) minus 1205731
1 minus 1205731
≺
1 + 119911
1 minus 119911
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
≺
1
1 minus 119911
(53)
by using Lemma 4 we get
(
Φ1 (119911) minus 1205731
1 minus 1205731
) lowast (
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
) ≺
1 + 119911
1 minus 119911
(54)
A simple calculation gives that
Φ1 (119911) lowast Φ2 (119911) ≺ 1 plusmn 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(55)
Therefore by the Lindeloff principle of subordinationwehave
R Φ1 (119911) lowast Φ2 (119911)
gt 1205730 = 1 minus 2 (1 minus 1205731) (1 minus 1205732) (119911 isin U) (56)
Since
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= Φ0 (119911)
(57)
by using (52) in conjunction with (56) and Lemma 4 we getthe following
R (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= R Φ0 (119911)
=
119901
119905
int
1
0
119906
(119901119905)minus1R (Φ1 lowast Φ2) (119906119911) 119889119906
ge
119901
119905
int
1
0
119906
(119901119905)minus1(1205730 + (1 minus 1205730)
1 minus 119906 |119911|
1 + 119906 |119911|
) 119889119906
gt 1205730 +
119901 (1 minus 1205730)
119905
int
1
0
119906
(119901119905)minus1 1 minus 119906
1 + 119906
119889119906
= 1 minus 4 (1 minus 1205731) (1 minus 1205732) (1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
= 120573 (119911 isin U)
(58)
This proves that119891(119911) isin H119899119898119901(120582 119905 ℎ) where the function ℎ(119911)
is given by (46)
6 Journal of Complex Analysis
In order to show that the value of 120573 is the least possiblewe take the functions 119891119895(119911) isin A119901 (119895 = 1 2) defined by
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1
times (120573119895 + (1 minus 120573119895)1 + 120585
1 minus 120585
) 119889120585
(59)
for which we have
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891119895 (119911))
1015840
= 120573119895 + (1 minus 120573119895)1 + 119911
1 minus 119911
(119895 = 1 2)
(60)
(Φ1 lowast Φ2) (119911) = 1 + 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(61)
Hence for 119891 isin A119901 given by (45) we obtain
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
=
119901
119905
int
1
0
119906
(119901119905)minus1(1 + 4 (1 minus 1205731) (1 minus 1205732)
119906119911
1 minus 119906119911
) 119889119906
997888rarr 120573 (as 119911 997888rarr minus1)
(62)
Finally for the case 119905 = 0 the proof of Theorem 11 is simpleso we choose to omit the details involved
4 Properties of the Libera-LivingstonTransform
For the function 119891 isin A119901 the functionF defined by
F (119911) =
120583 + 119901
119911
120583int
119911
0
120585
120583minus1119891 (120585) 119889120585 (R (120583) gt minus119901 119911 isin U)
(63)
is popularly known as the Libera-Livingston transform of 119891We state and prove the following
Theorem 12 Let 119891 isin H119899119898119901(120582 119905 ℎ) Then the function F
defined by (63) is in the classH119899119898119901(120582 119905
ℎ) where
ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
120585
120583+119901minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(64)
ConsequentlyF isin H119899119898119901(120582 119905 ℎ) The function ℎ(119911) is the best
dominant in (64)
Proof We define the function119870 on U by
119911
119901119870 (119911) = (1 minus 119905)D
(120582119901)
119905 (119899119898)F (119911)
+
119905
119901
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
(65)
Differentiating both the sides of (65) with respect to 119911 we get
119901119870 (119911) + 119911119870
1015840(119911)
= 119901 (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)
+ 119905119911
minus119901+1(D(120582119901)
119905 (119899119898)
119911F1015840(119911)
119901
)
1015840
(66)
Also the defining relation (63) yields
(120583 + 119901) 119891 (119911) = 120583F (119911) + 119911F1015840(119911) (67)
Now a routine calculation using (65) (66) and (67) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
(119911 isin U)
(68)
Since 119891 isin H119899119898119901(120582 119905 ℎ) we get the following from the
preceding equation (68)
119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
≺ ℎ (119911) (69)
Therefore by applying Lemma 2 we have
119870 (119911) ≺ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
119905
120583+119901minus1ℎ (119905) 119889119905 ≺ ℎ (119911)
(70)
This last subordination (70) is equivalent to
F isin H119899119898
119901(120582 119905
ℎ) sub H
119899119898
119901(120582 119905 ℎ) (71)
The proof of Theorem 12 is completed
Theorem 13 Let 119891 isin A119901 and suppose that the function F isdefined as in (63) If
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (120574 gt 0)
(72)
thenF isin H119899119898119901(120582 0
ℎ) where
ℎ (119911) =
120583 + 119901
120574
119911
minus(120583+119901)120574
times int
119911
0
120585
((120583+119901)120574)minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (R (120583) gt minus119901)
(73)
The function ℎ(119911) is the best dominant in (73)
Journal of Complex Analysis 7
Proof We define the function119867 on U by
119867(119911) = 119911
minus119901D(120582119901)
119905 (119899119898)F (119911) (74)
Differentiation of both the sides of (74) combined with theidentity
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
119901
= D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)(75)
gives
119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
) = 119867 (119911) +
119911119867
1015840(119911)
119901
(76)
By making use of (67) we simplify the subordinate of (72) asfollows
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus 120574)119867 (119911) + 120574119911
minus119901D(120582119901)
119905 (119899119898)
times (
120583
120583 + 119901
F (119911) +
119901
120583 + 119901
119911F1015840 (119911)
119901
)
= (1 minus 120574)119867 (119911)
120583120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+
119901120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (
119911F1015840 (119911)
119901
)
(77)
Next by using (74) and (76) the above identity furthersimplifies to the following
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus
119901120574
120583 + 119901
)119867 (119911) +
119901120574
120583 + 119901
(119867 (119911) +
119911119867
1015840(119911)
119901
)
= 119867 (119911) +
120574
120583 + 119901
119911119867
1015840(119911)
(78)
The subordination (72) is thus equivalent to
119867(119911) +
120574
120583 + 119901
119911119867
1015840(119911) ≺ ℎ (119911)
(R (120583) gt minus119901 120574 gt 0 119911 isin U)
(79)
Therefore an application of Lemma 2 yields the assertionof Theorem 13 The proof of Theorem 13 is completed
5 Inclusion Theorems
Theorem 14 Let 119905 gt 0 120574 gt 0 and 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574)
If 120574 le 1205740 then 119891(119911) isin H119899119898119901(120582 0 ℎ) where
1205740 =1
2
(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
minus1
(80)
The bound 1205740 is sharp when ℎ(119911) = 1(1 minus 119911)
Proof Let 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) By setting
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) (119911 isin U) (81)
we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119892 (119911) +
119905
119901
119911119892
1015840(119911)
(82)
Therefore by using Definition 1 we get
119892 (119911) +
119905
119901
119911119892
1015840(119911) ≺ 120574ℎ (119911) + 1 minus 120574 (83)
and an application of Lemma 2 yields
119892 (119911) ≺
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1ℎ (120585) 119889120585 + 1 minus 120574 = (ℎ lowast 120595) (119911)
(84)
where
120595 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574 (85)
It follows from (85) that if 0 lt 120574 le 1205740 where 1205740 gt 1 is givenby (80) then
R (120595 (119911)) =
120574119901
119905
int
1
0
119906
(119901119905)minus1R(
1
1 minus 119906119911
) 119889119906 + 1 minus 120574
gt
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906 + 1 minus 120574
ge
1
2
(119911 isin U)
(86)
Equivalently
120595 (119911) ≺
1
1 minus 119911
(119911 isin U) (87)
Since ℎ(119911) and 120595(119911) are both convex univalent functions inU using Lemma 4 we obtain from (84) that
119892 (119911) ≺ (ℎ lowast 120595) (119911) ≺ ℎ (119911) (88)
Therefore in view of (81) we have
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (89)
Or equivalently 119891(119911) isin H119899119898119901(120582 0 ℎ)
In order to prove that the bound on 1205740 is the best possibleset ℎ(119911) = 1(1 minus 119911) and let the function 119891 be defined onU by
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574
(90)
8 Journal of Complex Analysis
It can be readily verified that 119891 isin A119901 and
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 120574ℎ (119911) + 1 minus 120574
(91)
Thus 119891(119911) isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) Also for 120574 gt 1205740 and
119911 rarr minus1 we have
R 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) 997888rarr
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906
+ 1 minus 120574 lt
1
2
(92)
which shows that
119891 (119911) notin H119899119898
119901(120582 0 ℎ) (93)
Hence the bound 1205740 cannot be increased when ℎ(119911) = 1(1 minus119911) The proof of Theorem 14 is completed
Taking119898 = 0 we use the notation
H1198990
119901(120582 119905 ℎ) = H
119899
119901(120582 119905 ℎ) (94)
in the following theorem
Theorem 15 Let 0 le 1199051 lt 1199052 Then
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (95)
Proof Let 0 le 1199051 lt 1199052 and suppose that 119891 isin H119899119901(120582 1199052 ℎ) We
define
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899 0) 119891 (119911) (119911 isin U) (96)
The function 119892 is analytic in U with 119892(0) = 1 Differenti-ating the expressions on both the sides of (96) with respect to119911 and using Definition 1 we have
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
= 119892 (119911) +
1199052
119901
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U)
(97)
Hence an application of Lemma 2 yields
119892 (119911) ≺ ℎ (119911) (119911 isin U) (98)
Noting that 0 le 11990511199052 lt 1 and ℎ(119911) is convex univalent in Uit follows from (96) (97) and (98) that
(1 minus 1199051) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199051
119911
minus119901+1(D(120582119901)
119905 (119899 0)119891(119911))
1015840
=
1199051
1199052
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
+ (1 minus
1199051
1199052
)119892 (119911)
≺ ℎ (119911) (119911 isin U)
(99)
Thus 119891 isin H119899119901(120582 1199051 ℎ) Consequently
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (100)
The proof of Theorem 15 is completed
Remark 16 The particular case 119899 = 1 of Theorem 15 gives aresult of Liu ([17] Theorem 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Patel and A K Mishra ldquoOn certain subclasses of multivalentfunctions associated with an extended fractional differintegraloperatorrdquo Journal of Mathematical Analysis and Applicationsvol 332 no 1 pp 109ndash122 2007
[2] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987
[3] H M Srivastava and S Owa ldquoAn application of the fractionalderivativerdquo Mathematica Japonica vol 29 no 3 pp 383ndash3891984
[4] A K Mishra and P Gochhayat ldquoApplications of the Owa-Srivastava operator to the class of 119896-uniformly convex func-tionsrdquo Fractional Calculus amp Applied Analysis vol 9 no 4 pp323ndash331 2006
[5] A K Mishra and P Gochhayat ldquoThe Fekete-Szego problem fork-uniformly convex functions and for a class defined by theOwa-Srivastava operatorrdquo Journal ofMathematical Analysis andApplications vol 347 no 2 pp 563ndash572 2008
[6] H M Srivastava A K Mishra and S N Kund ldquoCertainclasses of analytic functions associated with iterations of theOwa-Srivastava fractional derivative operatorrdquo Southeast AsianBulletin of Mathematics vol 37 no 3 pp 413ndash435 2013
[7] HM Srivastava andMK Aouf ldquoA certain fractional derivativeoperator and its applications to a new class of analytic andmultivalent functions with negative coefficientsrdquo Journal ofMathematical Analysis and Applications vol 171 no 1 pp 1ndash131992
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Complex Analysis
Proof For every 119891 and 119892 inA119901 we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
= (1 minus 119905) (119911
minus119901119892 (119911)) lowast (119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911))
+
119905
119901
(119911
minus119901119892 (119911)) lowast (119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
)
= (119911
minus119901119892 (119911)) lowast Ψ (119911)
(29)
where
Ψ (119911) = (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
(30)
Now if 119891 isin H119899119898119901(120582 119905 ℎ) then
Ψ (119911) ≺ ℎ (119911) (119911 isin U) (31)
Furthermore condition (27) is equivalent to
119911
minus119901119892 (119911) ≺
1
1 minus 119911
(119911 isin U) (32)
Therefore an application of Lemma 4 in (29) yields
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
≺
1
1 minus 119911
lowast ℎ (119911)
= ℎ (119911)
(33)
This shows that 119891 lowast 119892 isin H119899119898119901(120582 119905 ℎ) The proof of
Theorem 6 is completed
Remark 7 Taking 119899 = 1 119898 = 0 inTheorem 6 we get a recentresult of Liu ([17]Theorem 3)The choices 119899 = 1 119898 = 0 119901 =1 120582 = 1 and ℎ(119911) = (1 + 119886119911)(1 + 119887119911) (minus1 le 119887 lt 1 119886 gt 119887)
yield a result of Yang ([22] Theorem 4)
Corollary 8 Let the function 119891 given by (2) be a member ofH119899119898119901(120582 119905 ℎ) and
119904119903 (119911) = 119911119901+
119903minus1
sum
119896=1
119886119896119911119896+119901
(119903 isin N 1 119911 isin U) (34)
Then the function
120590119903 (119911) = int
1
0
119905
minus119901119904119903 (119905119911) 119889119905
(35)
is in the classH119899119898119901(120582 119905 ℎ)
Proof Let 119891 isin H119899119898119901(120582 119905 ℎ) We note that
120590119903 (119911) = 119911119901+
119903minus1
sum
119896=1
119886119896
119896 + 1
119911
119896+119901= (119891 lowast 119892119903) (119911) (119903 isin N 1)
(36)
where
119891 (119911) = 119911
119901+
infin
sum
119896=1
119886119896119911119896+119901
119892119903 (119911) = 119911
119901+
119903minus1
sum
119896=1
119911
119896+119901
119896 + 1
isin A119901
(37)
Also for 119903 isin N 1 it is well known [18] that
R 119911
minus119901119892119903 (119911) gt
1
2
(119911 isin U) (38)
In view of (36) and (38) an application of Theorem 6 gives
120590119903 isin H119899119898
119901(120582 119905 ℎ) (39)
The proof of Corollary 8 is completed
Theorem 9 Let the function 119892 in A119901 be such that 119911minus119901+1119892(119911)is a prestarlike function of order 120588 (120588 lt 1) If119891 isin H119899119898
119901(120582 119905 ℎ)
then119891 lowast 119892 isin H
119899119898
119901(120582 119905 ℎ) (40)
Proof Let 119891 isin H119899119898119901(120582 119905 ℎ) and 119892 isin A119901 Then (29) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911))
1015840
=
(119911
minus119901+1119892 (119911)) lowast (119911Ψ (119911))
(119911
minus119901+1119892 (119911)) lowast 119911
(119911 isin U)
(41)
where Ψ(119911) is defined as in (30) We noted in the proof ofTheorem 6 thatΨ(119911) ≺ ℎ(119911) Since 119911minus119901+1119892(119911) isin PSlowast(120588) 119911 isinSlowast(120588) and ℎ(119911) is convex univalent in U an application ofLemma 3 in (41) yields the following
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) (119891 lowast 119892) (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)(119891 lowast 119892)(119911))
1015840
≺ ℎ (119911) (119911 isin U)
(42)
Therefore 119891 lowast 119892 isin H119899119898119901(120582 119905 ℎ) The proof Theorem 9 is
completed
Taking 120588 = 12 in Theorem 9 we get the following
Corollary 10 Let 119891 isin H119899119898119901(120582 119905 ℎ) and suppose that 119892 inA119901
is such that 119911minus119901+1119892 isin Slowast(12) Then
119891 lowast 119892 isin H119899119898
119901(120582 119905 ℎ) (43)
In particular if 119911minus119901+1119892 is univalent convex then 119891 lowast 119892 isin
H119899119898119901(120582 119905 ℎ)
Journal of Complex Analysis 5
In the following theorem we discuss convolution proper-ties of the function classH(120582 119905 ℎ)when ℎ is a right half planemapping
Theorem 11 Let 119905 ge 0 and suppose that each of the functions119891119895 (119895 = 1 2) is a member of the classH119899119898
119901(120582 119905 ℎ119895) where
ℎ119895 (119911) =
1 + (1 minus 2120573119895) 119911
1 minus 119911
(0 le 120573119895 lt 1) (44)
If 119891 isin A119901 is defined by the following
119891 (119911) = D(120582119901)
119905 (119899119898) (1198911 lowast 1198912) (119911) (119911 isin U) (45)
then 119891 isin H119899119898119901(120582 119905 ℎ) where
ℎ (119911) =
1 + (1 minus 2120573) 119911
1 minus 119911
(46)
and 120573 is given by
120573 =
1 minus 4 (1 minus 1205731) (1 minus 1205732)
times(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906) (120582 gt 0)
1 minus 2 (1 minus 1205731) (1 minus 1205732) (120582 = 0)
(47)
The bound on 120573 is the best possible
Proof We consider the case 119905 gt 0 Suppose that 119891119895 isin
H119899119898119901(120582 119905 ℎ119895) (119895 = 1 2) where ℎ119895(119911) is given by (44) By
setting
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891119895(119911))
1015840(48)
we see in the light of Definition 1 that
Φ119895 (119911) ≺ ℎ119895 (119911) (119895 = 1 2) (49)
A routine calculation yields the following
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1Φ119895 (120585) 119889120585
(119895 = 1 2)
(50)
Now if 119891(119911) is defined by (45) then using (50) we get that
D(120582119901)
119905 (119899119898) 119891 (119911)
= D(120582119901)
119905 (119899119898) 1198911 (119911) lowastD(120582119901)
119905 (119899119898) 1198912 (119911)
= (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ1 (119906119911) 119889119906)
lowast (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ2 (119906119911) 119889119906)
=
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ0 (119906119911) 119889119906
(51)
where
Φ0 (119911) =
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1(Φ1 lowast Φ2) (119906119911) 119889119906
(52)
Since
Φ1 (119911) minus 1205731
1 minus 1205731
≺
1 + 119911
1 minus 119911
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
≺
1
1 minus 119911
(53)
by using Lemma 4 we get
(
Φ1 (119911) minus 1205731
1 minus 1205731
) lowast (
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
) ≺
1 + 119911
1 minus 119911
(54)
A simple calculation gives that
Φ1 (119911) lowast Φ2 (119911) ≺ 1 plusmn 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(55)
Therefore by the Lindeloff principle of subordinationwehave
R Φ1 (119911) lowast Φ2 (119911)
gt 1205730 = 1 minus 2 (1 minus 1205731) (1 minus 1205732) (119911 isin U) (56)
Since
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= Φ0 (119911)
(57)
by using (52) in conjunction with (56) and Lemma 4 we getthe following
R (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= R Φ0 (119911)
=
119901
119905
int
1
0
119906
(119901119905)minus1R (Φ1 lowast Φ2) (119906119911) 119889119906
ge
119901
119905
int
1
0
119906
(119901119905)minus1(1205730 + (1 minus 1205730)
1 minus 119906 |119911|
1 + 119906 |119911|
) 119889119906
gt 1205730 +
119901 (1 minus 1205730)
119905
int
1
0
119906
(119901119905)minus1 1 minus 119906
1 + 119906
119889119906
= 1 minus 4 (1 minus 1205731) (1 minus 1205732) (1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
= 120573 (119911 isin U)
(58)
This proves that119891(119911) isin H119899119898119901(120582 119905 ℎ) where the function ℎ(119911)
is given by (46)
6 Journal of Complex Analysis
In order to show that the value of 120573 is the least possiblewe take the functions 119891119895(119911) isin A119901 (119895 = 1 2) defined by
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1
times (120573119895 + (1 minus 120573119895)1 + 120585
1 minus 120585
) 119889120585
(59)
for which we have
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891119895 (119911))
1015840
= 120573119895 + (1 minus 120573119895)1 + 119911
1 minus 119911
(119895 = 1 2)
(60)
(Φ1 lowast Φ2) (119911) = 1 + 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(61)
Hence for 119891 isin A119901 given by (45) we obtain
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
=
119901
119905
int
1
0
119906
(119901119905)minus1(1 + 4 (1 minus 1205731) (1 minus 1205732)
119906119911
1 minus 119906119911
) 119889119906
997888rarr 120573 (as 119911 997888rarr minus1)
(62)
Finally for the case 119905 = 0 the proof of Theorem 11 is simpleso we choose to omit the details involved
4 Properties of the Libera-LivingstonTransform
For the function 119891 isin A119901 the functionF defined by
F (119911) =
120583 + 119901
119911
120583int
119911
0
120585
120583minus1119891 (120585) 119889120585 (R (120583) gt minus119901 119911 isin U)
(63)
is popularly known as the Libera-Livingston transform of 119891We state and prove the following
Theorem 12 Let 119891 isin H119899119898119901(120582 119905 ℎ) Then the function F
defined by (63) is in the classH119899119898119901(120582 119905
ℎ) where
ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
120585
120583+119901minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(64)
ConsequentlyF isin H119899119898119901(120582 119905 ℎ) The function ℎ(119911) is the best
dominant in (64)
Proof We define the function119870 on U by
119911
119901119870 (119911) = (1 minus 119905)D
(120582119901)
119905 (119899119898)F (119911)
+
119905
119901
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
(65)
Differentiating both the sides of (65) with respect to 119911 we get
119901119870 (119911) + 119911119870
1015840(119911)
= 119901 (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)
+ 119905119911
minus119901+1(D(120582119901)
119905 (119899119898)
119911F1015840(119911)
119901
)
1015840
(66)
Also the defining relation (63) yields
(120583 + 119901) 119891 (119911) = 120583F (119911) + 119911F1015840(119911) (67)
Now a routine calculation using (65) (66) and (67) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
(119911 isin U)
(68)
Since 119891 isin H119899119898119901(120582 119905 ℎ) we get the following from the
preceding equation (68)
119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
≺ ℎ (119911) (69)
Therefore by applying Lemma 2 we have
119870 (119911) ≺ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
119905
120583+119901minus1ℎ (119905) 119889119905 ≺ ℎ (119911)
(70)
This last subordination (70) is equivalent to
F isin H119899119898
119901(120582 119905
ℎ) sub H
119899119898
119901(120582 119905 ℎ) (71)
The proof of Theorem 12 is completed
Theorem 13 Let 119891 isin A119901 and suppose that the function F isdefined as in (63) If
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (120574 gt 0)
(72)
thenF isin H119899119898119901(120582 0
ℎ) where
ℎ (119911) =
120583 + 119901
120574
119911
minus(120583+119901)120574
times int
119911
0
120585
((120583+119901)120574)minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (R (120583) gt minus119901)
(73)
The function ℎ(119911) is the best dominant in (73)
Journal of Complex Analysis 7
Proof We define the function119867 on U by
119867(119911) = 119911
minus119901D(120582119901)
119905 (119899119898)F (119911) (74)
Differentiation of both the sides of (74) combined with theidentity
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
119901
= D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)(75)
gives
119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
) = 119867 (119911) +
119911119867
1015840(119911)
119901
(76)
By making use of (67) we simplify the subordinate of (72) asfollows
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus 120574)119867 (119911) + 120574119911
minus119901D(120582119901)
119905 (119899119898)
times (
120583
120583 + 119901
F (119911) +
119901
120583 + 119901
119911F1015840 (119911)
119901
)
= (1 minus 120574)119867 (119911)
120583120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+
119901120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (
119911F1015840 (119911)
119901
)
(77)
Next by using (74) and (76) the above identity furthersimplifies to the following
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus
119901120574
120583 + 119901
)119867 (119911) +
119901120574
120583 + 119901
(119867 (119911) +
119911119867
1015840(119911)
119901
)
= 119867 (119911) +
120574
120583 + 119901
119911119867
1015840(119911)
(78)
The subordination (72) is thus equivalent to
119867(119911) +
120574
120583 + 119901
119911119867
1015840(119911) ≺ ℎ (119911)
(R (120583) gt minus119901 120574 gt 0 119911 isin U)
(79)
Therefore an application of Lemma 2 yields the assertionof Theorem 13 The proof of Theorem 13 is completed
5 Inclusion Theorems
Theorem 14 Let 119905 gt 0 120574 gt 0 and 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574)
If 120574 le 1205740 then 119891(119911) isin H119899119898119901(120582 0 ℎ) where
1205740 =1
2
(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
minus1
(80)
The bound 1205740 is sharp when ℎ(119911) = 1(1 minus 119911)
Proof Let 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) By setting
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) (119911 isin U) (81)
we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119892 (119911) +
119905
119901
119911119892
1015840(119911)
(82)
Therefore by using Definition 1 we get
119892 (119911) +
119905
119901
119911119892
1015840(119911) ≺ 120574ℎ (119911) + 1 minus 120574 (83)
and an application of Lemma 2 yields
119892 (119911) ≺
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1ℎ (120585) 119889120585 + 1 minus 120574 = (ℎ lowast 120595) (119911)
(84)
where
120595 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574 (85)
It follows from (85) that if 0 lt 120574 le 1205740 where 1205740 gt 1 is givenby (80) then
R (120595 (119911)) =
120574119901
119905
int
1
0
119906
(119901119905)minus1R(
1
1 minus 119906119911
) 119889119906 + 1 minus 120574
gt
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906 + 1 minus 120574
ge
1
2
(119911 isin U)
(86)
Equivalently
120595 (119911) ≺
1
1 minus 119911
(119911 isin U) (87)
Since ℎ(119911) and 120595(119911) are both convex univalent functions inU using Lemma 4 we obtain from (84) that
119892 (119911) ≺ (ℎ lowast 120595) (119911) ≺ ℎ (119911) (88)
Therefore in view of (81) we have
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (89)
Or equivalently 119891(119911) isin H119899119898119901(120582 0 ℎ)
In order to prove that the bound on 1205740 is the best possibleset ℎ(119911) = 1(1 minus 119911) and let the function 119891 be defined onU by
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574
(90)
8 Journal of Complex Analysis
It can be readily verified that 119891 isin A119901 and
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 120574ℎ (119911) + 1 minus 120574
(91)
Thus 119891(119911) isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) Also for 120574 gt 1205740 and
119911 rarr minus1 we have
R 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) 997888rarr
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906
+ 1 minus 120574 lt
1
2
(92)
which shows that
119891 (119911) notin H119899119898
119901(120582 0 ℎ) (93)
Hence the bound 1205740 cannot be increased when ℎ(119911) = 1(1 minus119911) The proof of Theorem 14 is completed
Taking119898 = 0 we use the notation
H1198990
119901(120582 119905 ℎ) = H
119899
119901(120582 119905 ℎ) (94)
in the following theorem
Theorem 15 Let 0 le 1199051 lt 1199052 Then
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (95)
Proof Let 0 le 1199051 lt 1199052 and suppose that 119891 isin H119899119901(120582 1199052 ℎ) We
define
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899 0) 119891 (119911) (119911 isin U) (96)
The function 119892 is analytic in U with 119892(0) = 1 Differenti-ating the expressions on both the sides of (96) with respect to119911 and using Definition 1 we have
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
= 119892 (119911) +
1199052
119901
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U)
(97)
Hence an application of Lemma 2 yields
119892 (119911) ≺ ℎ (119911) (119911 isin U) (98)
Noting that 0 le 11990511199052 lt 1 and ℎ(119911) is convex univalent in Uit follows from (96) (97) and (98) that
(1 minus 1199051) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199051
119911
minus119901+1(D(120582119901)
119905 (119899 0)119891(119911))
1015840
=
1199051
1199052
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
+ (1 minus
1199051
1199052
)119892 (119911)
≺ ℎ (119911) (119911 isin U)
(99)
Thus 119891 isin H119899119901(120582 1199051 ℎ) Consequently
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (100)
The proof of Theorem 15 is completed
Remark 16 The particular case 119899 = 1 of Theorem 15 gives aresult of Liu ([17] Theorem 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Patel and A K Mishra ldquoOn certain subclasses of multivalentfunctions associated with an extended fractional differintegraloperatorrdquo Journal of Mathematical Analysis and Applicationsvol 332 no 1 pp 109ndash122 2007
[2] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987
[3] H M Srivastava and S Owa ldquoAn application of the fractionalderivativerdquo Mathematica Japonica vol 29 no 3 pp 383ndash3891984
[4] A K Mishra and P Gochhayat ldquoApplications of the Owa-Srivastava operator to the class of 119896-uniformly convex func-tionsrdquo Fractional Calculus amp Applied Analysis vol 9 no 4 pp323ndash331 2006
[5] A K Mishra and P Gochhayat ldquoThe Fekete-Szego problem fork-uniformly convex functions and for a class defined by theOwa-Srivastava operatorrdquo Journal ofMathematical Analysis andApplications vol 347 no 2 pp 563ndash572 2008
[6] H M Srivastava A K Mishra and S N Kund ldquoCertainclasses of analytic functions associated with iterations of theOwa-Srivastava fractional derivative operatorrdquo Southeast AsianBulletin of Mathematics vol 37 no 3 pp 413ndash435 2013
[7] HM Srivastava andMK Aouf ldquoA certain fractional derivativeoperator and its applications to a new class of analytic andmultivalent functions with negative coefficientsrdquo Journal ofMathematical Analysis and Applications vol 171 no 1 pp 1ndash131992
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Complex Analysis 5
In the following theorem we discuss convolution proper-ties of the function classH(120582 119905 ℎ)when ℎ is a right half planemapping
Theorem 11 Let 119905 ge 0 and suppose that each of the functions119891119895 (119895 = 1 2) is a member of the classH119899119898
119901(120582 119905 ℎ119895) where
ℎ119895 (119911) =
1 + (1 minus 2120573119895) 119911
1 minus 119911
(0 le 120573119895 lt 1) (44)
If 119891 isin A119901 is defined by the following
119891 (119911) = D(120582119901)
119905 (119899119898) (1198911 lowast 1198912) (119911) (119911 isin U) (45)
then 119891 isin H119899119898119901(120582 119905 ℎ) where
ℎ (119911) =
1 + (1 minus 2120573) 119911
1 minus 119911
(46)
and 120573 is given by
120573 =
1 minus 4 (1 minus 1205731) (1 minus 1205732)
times(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906) (120582 gt 0)
1 minus 2 (1 minus 1205731) (1 minus 1205732) (120582 = 0)
(47)
The bound on 120573 is the best possible
Proof We consider the case 119905 gt 0 Suppose that 119891119895 isin
H119899119898119901(120582 119905 ℎ119895) (119895 = 1 2) where ℎ119895(119911) is given by (44) By
setting
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891119895(119911))
1015840(48)
we see in the light of Definition 1 that
Φ119895 (119911) ≺ ℎ119895 (119911) (119895 = 1 2) (49)
A routine calculation yields the following
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1Φ119895 (120585) 119889120585
(119895 = 1 2)
(50)
Now if 119891(119911) is defined by (45) then using (50) we get that
D(120582119901)
119905 (119899119898) 119891 (119911)
= D(120582119901)
119905 (119899119898) 1198911 (119911) lowastD(120582119901)
119905 (119899119898) 1198912 (119911)
= (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ1 (119906119911) 119889119906)
lowast (
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ2 (119906119911) 119889119906)
=
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1Φ0 (119906119911) 119889119906
(51)
where
Φ0 (119911) =
119901
119905
119911
119901int
1
0
119906
(119901119905)minus1(Φ1 lowast Φ2) (119906119911) 119889119906
(52)
Since
Φ1 (119911) minus 1205731
1 minus 1205731
≺
1 + 119911
1 minus 119911
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
≺
1
1 minus 119911
(53)
by using Lemma 4 we get
(
Φ1 (119911) minus 1205731
1 minus 1205731
) lowast (
1
2
+
Φ2 (119911) minus 1205732
2 (1 minus 1205732)
) ≺
1 + 119911
1 minus 119911
(54)
A simple calculation gives that
Φ1 (119911) lowast Φ2 (119911) ≺ 1 plusmn 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(55)
Therefore by the Lindeloff principle of subordinationwehave
R Φ1 (119911) lowast Φ2 (119911)
gt 1205730 = 1 minus 2 (1 minus 1205731) (1 minus 1205732) (119911 isin U) (56)
Since
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= Φ0 (119911)
(57)
by using (52) in conjunction with (56) and Lemma 4 we getthe following
R (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898)119891(119911))
1015840
= R Φ0 (119911)
=
119901
119905
int
1
0
119906
(119901119905)minus1R (Φ1 lowast Φ2) (119906119911) 119889119906
ge
119901
119905
int
1
0
119906
(119901119905)minus1(1205730 + (1 minus 1205730)
1 minus 119906 |119911|
1 + 119906 |119911|
) 119889119906
gt 1205730 +
119901 (1 minus 1205730)
119905
int
1
0
119906
(119901119905)minus1 1 minus 119906
1 + 119906
119889119906
= 1 minus 4 (1 minus 1205731) (1 minus 1205732) (1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
= 120573 (119911 isin U)
(58)
This proves that119891(119911) isin H119899119898119901(120582 119905 ℎ) where the function ℎ(119911)
is given by (46)
6 Journal of Complex Analysis
In order to show that the value of 120573 is the least possiblewe take the functions 119891119895(119911) isin A119901 (119895 = 1 2) defined by
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1
times (120573119895 + (1 minus 120573119895)1 + 120585
1 minus 120585
) 119889120585
(59)
for which we have
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891119895 (119911))
1015840
= 120573119895 + (1 minus 120573119895)1 + 119911
1 minus 119911
(119895 = 1 2)
(60)
(Φ1 lowast Φ2) (119911) = 1 + 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(61)
Hence for 119891 isin A119901 given by (45) we obtain
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
=
119901
119905
int
1
0
119906
(119901119905)minus1(1 + 4 (1 minus 1205731) (1 minus 1205732)
119906119911
1 minus 119906119911
) 119889119906
997888rarr 120573 (as 119911 997888rarr minus1)
(62)
Finally for the case 119905 = 0 the proof of Theorem 11 is simpleso we choose to omit the details involved
4 Properties of the Libera-LivingstonTransform
For the function 119891 isin A119901 the functionF defined by
F (119911) =
120583 + 119901
119911
120583int
119911
0
120585
120583minus1119891 (120585) 119889120585 (R (120583) gt minus119901 119911 isin U)
(63)
is popularly known as the Libera-Livingston transform of 119891We state and prove the following
Theorem 12 Let 119891 isin H119899119898119901(120582 119905 ℎ) Then the function F
defined by (63) is in the classH119899119898119901(120582 119905
ℎ) where
ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
120585
120583+119901minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(64)
ConsequentlyF isin H119899119898119901(120582 119905 ℎ) The function ℎ(119911) is the best
dominant in (64)
Proof We define the function119870 on U by
119911
119901119870 (119911) = (1 minus 119905)D
(120582119901)
119905 (119899119898)F (119911)
+
119905
119901
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
(65)
Differentiating both the sides of (65) with respect to 119911 we get
119901119870 (119911) + 119911119870
1015840(119911)
= 119901 (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)
+ 119905119911
minus119901+1(D(120582119901)
119905 (119899119898)
119911F1015840(119911)
119901
)
1015840
(66)
Also the defining relation (63) yields
(120583 + 119901) 119891 (119911) = 120583F (119911) + 119911F1015840(119911) (67)
Now a routine calculation using (65) (66) and (67) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
(119911 isin U)
(68)
Since 119891 isin H119899119898119901(120582 119905 ℎ) we get the following from the
preceding equation (68)
119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
≺ ℎ (119911) (69)
Therefore by applying Lemma 2 we have
119870 (119911) ≺ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
119905
120583+119901minus1ℎ (119905) 119889119905 ≺ ℎ (119911)
(70)
This last subordination (70) is equivalent to
F isin H119899119898
119901(120582 119905
ℎ) sub H
119899119898
119901(120582 119905 ℎ) (71)
The proof of Theorem 12 is completed
Theorem 13 Let 119891 isin A119901 and suppose that the function F isdefined as in (63) If
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (120574 gt 0)
(72)
thenF isin H119899119898119901(120582 0
ℎ) where
ℎ (119911) =
120583 + 119901
120574
119911
minus(120583+119901)120574
times int
119911
0
120585
((120583+119901)120574)minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (R (120583) gt minus119901)
(73)
The function ℎ(119911) is the best dominant in (73)
Journal of Complex Analysis 7
Proof We define the function119867 on U by
119867(119911) = 119911
minus119901D(120582119901)
119905 (119899119898)F (119911) (74)
Differentiation of both the sides of (74) combined with theidentity
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
119901
= D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)(75)
gives
119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
) = 119867 (119911) +
119911119867
1015840(119911)
119901
(76)
By making use of (67) we simplify the subordinate of (72) asfollows
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus 120574)119867 (119911) + 120574119911
minus119901D(120582119901)
119905 (119899119898)
times (
120583
120583 + 119901
F (119911) +
119901
120583 + 119901
119911F1015840 (119911)
119901
)
= (1 minus 120574)119867 (119911)
120583120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+
119901120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (
119911F1015840 (119911)
119901
)
(77)
Next by using (74) and (76) the above identity furthersimplifies to the following
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus
119901120574
120583 + 119901
)119867 (119911) +
119901120574
120583 + 119901
(119867 (119911) +
119911119867
1015840(119911)
119901
)
= 119867 (119911) +
120574
120583 + 119901
119911119867
1015840(119911)
(78)
The subordination (72) is thus equivalent to
119867(119911) +
120574
120583 + 119901
119911119867
1015840(119911) ≺ ℎ (119911)
(R (120583) gt minus119901 120574 gt 0 119911 isin U)
(79)
Therefore an application of Lemma 2 yields the assertionof Theorem 13 The proof of Theorem 13 is completed
5 Inclusion Theorems
Theorem 14 Let 119905 gt 0 120574 gt 0 and 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574)
If 120574 le 1205740 then 119891(119911) isin H119899119898119901(120582 0 ℎ) where
1205740 =1
2
(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
minus1
(80)
The bound 1205740 is sharp when ℎ(119911) = 1(1 minus 119911)
Proof Let 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) By setting
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) (119911 isin U) (81)
we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119892 (119911) +
119905
119901
119911119892
1015840(119911)
(82)
Therefore by using Definition 1 we get
119892 (119911) +
119905
119901
119911119892
1015840(119911) ≺ 120574ℎ (119911) + 1 minus 120574 (83)
and an application of Lemma 2 yields
119892 (119911) ≺
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1ℎ (120585) 119889120585 + 1 minus 120574 = (ℎ lowast 120595) (119911)
(84)
where
120595 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574 (85)
It follows from (85) that if 0 lt 120574 le 1205740 where 1205740 gt 1 is givenby (80) then
R (120595 (119911)) =
120574119901
119905
int
1
0
119906
(119901119905)minus1R(
1
1 minus 119906119911
) 119889119906 + 1 minus 120574
gt
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906 + 1 minus 120574
ge
1
2
(119911 isin U)
(86)
Equivalently
120595 (119911) ≺
1
1 minus 119911
(119911 isin U) (87)
Since ℎ(119911) and 120595(119911) are both convex univalent functions inU using Lemma 4 we obtain from (84) that
119892 (119911) ≺ (ℎ lowast 120595) (119911) ≺ ℎ (119911) (88)
Therefore in view of (81) we have
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (89)
Or equivalently 119891(119911) isin H119899119898119901(120582 0 ℎ)
In order to prove that the bound on 1205740 is the best possibleset ℎ(119911) = 1(1 minus 119911) and let the function 119891 be defined onU by
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574
(90)
8 Journal of Complex Analysis
It can be readily verified that 119891 isin A119901 and
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 120574ℎ (119911) + 1 minus 120574
(91)
Thus 119891(119911) isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) Also for 120574 gt 1205740 and
119911 rarr minus1 we have
R 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) 997888rarr
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906
+ 1 minus 120574 lt
1
2
(92)
which shows that
119891 (119911) notin H119899119898
119901(120582 0 ℎ) (93)
Hence the bound 1205740 cannot be increased when ℎ(119911) = 1(1 minus119911) The proof of Theorem 14 is completed
Taking119898 = 0 we use the notation
H1198990
119901(120582 119905 ℎ) = H
119899
119901(120582 119905 ℎ) (94)
in the following theorem
Theorem 15 Let 0 le 1199051 lt 1199052 Then
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (95)
Proof Let 0 le 1199051 lt 1199052 and suppose that 119891 isin H119899119901(120582 1199052 ℎ) We
define
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899 0) 119891 (119911) (119911 isin U) (96)
The function 119892 is analytic in U with 119892(0) = 1 Differenti-ating the expressions on both the sides of (96) with respect to119911 and using Definition 1 we have
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
= 119892 (119911) +
1199052
119901
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U)
(97)
Hence an application of Lemma 2 yields
119892 (119911) ≺ ℎ (119911) (119911 isin U) (98)
Noting that 0 le 11990511199052 lt 1 and ℎ(119911) is convex univalent in Uit follows from (96) (97) and (98) that
(1 minus 1199051) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199051
119911
minus119901+1(D(120582119901)
119905 (119899 0)119891(119911))
1015840
=
1199051
1199052
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
+ (1 minus
1199051
1199052
)119892 (119911)
≺ ℎ (119911) (119911 isin U)
(99)
Thus 119891 isin H119899119901(120582 1199051 ℎ) Consequently
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (100)
The proof of Theorem 15 is completed
Remark 16 The particular case 119899 = 1 of Theorem 15 gives aresult of Liu ([17] Theorem 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Patel and A K Mishra ldquoOn certain subclasses of multivalentfunctions associated with an extended fractional differintegraloperatorrdquo Journal of Mathematical Analysis and Applicationsvol 332 no 1 pp 109ndash122 2007
[2] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987
[3] H M Srivastava and S Owa ldquoAn application of the fractionalderivativerdquo Mathematica Japonica vol 29 no 3 pp 383ndash3891984
[4] A K Mishra and P Gochhayat ldquoApplications of the Owa-Srivastava operator to the class of 119896-uniformly convex func-tionsrdquo Fractional Calculus amp Applied Analysis vol 9 no 4 pp323ndash331 2006
[5] A K Mishra and P Gochhayat ldquoThe Fekete-Szego problem fork-uniformly convex functions and for a class defined by theOwa-Srivastava operatorrdquo Journal ofMathematical Analysis andApplications vol 347 no 2 pp 563ndash572 2008
[6] H M Srivastava A K Mishra and S N Kund ldquoCertainclasses of analytic functions associated with iterations of theOwa-Srivastava fractional derivative operatorrdquo Southeast AsianBulletin of Mathematics vol 37 no 3 pp 413ndash435 2013
[7] HM Srivastava andMK Aouf ldquoA certain fractional derivativeoperator and its applications to a new class of analytic andmultivalent functions with negative coefficientsrdquo Journal ofMathematical Analysis and Applications vol 171 no 1 pp 1ndash131992
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Complex Analysis
In order to show that the value of 120573 is the least possiblewe take the functions 119891119895(119911) isin A119901 (119895 = 1 2) defined by
D(120582119901)
119905 (119899119898) 119891119895 (119911) =
119901
119905
119911
minus119901(1minus119905)119905int
119911
0
120585
(119901119905)minus1
times (120573119895 + (1 minus 120573119895)1 + 120585
1 minus 120585
) 119889120585
(59)
for which we have
Φ119895 (119911) = (1 minus 119905) 119911minus119901D(120582119901)
119905 (119899119898) 119891119895 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891119895 (119911))
1015840
= 120573119895 + (1 minus 120573119895)1 + 119911
1 minus 119911
(119895 = 1 2)
(60)
(Φ1 lowast Φ2) (119911) = 1 + 4 (1 minus 1205731) (1 minus 1205732)119911
1 minus 119911
(61)
Hence for 119891 isin A119901 given by (45) we obtain
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119901
119905
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
=
119901
119905
int
1
0
119906
(119901119905)minus1(1 + 4 (1 minus 1205731) (1 minus 1205732)
119906119911
1 minus 119906119911
) 119889119906
997888rarr 120573 (as 119911 997888rarr minus1)
(62)
Finally for the case 119905 = 0 the proof of Theorem 11 is simpleso we choose to omit the details involved
4 Properties of the Libera-LivingstonTransform
For the function 119891 isin A119901 the functionF defined by
F (119911) =
120583 + 119901
119911
120583int
119911
0
120585
120583minus1119891 (120585) 119889120585 (R (120583) gt minus119901 119911 isin U)
(63)
is popularly known as the Libera-Livingston transform of 119891We state and prove the following
Theorem 12 Let 119891 isin H119899119898119901(120582 119905 ℎ) Then the function F
defined by (63) is in the classH119899119898119901(120582 119905
ℎ) where
ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
120585
120583+119901minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (119911 isin U)
(64)
ConsequentlyF isin H119899119898119901(120582 119905 ℎ) The function ℎ(119911) is the best
dominant in (64)
Proof We define the function119870 on U by
119911
119901119870 (119911) = (1 minus 119905)D
(120582119901)
119905 (119899119898)F (119911)
+
119905
119901
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
(65)
Differentiating both the sides of (65) with respect to 119911 we get
119901119870 (119911) + 119911119870
1015840(119911)
= 119901 (1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)
+ 119905119911
minus119901+1(D(120582119901)
119905 (119899119898)
119911F1015840(119911)
119901
)
1015840
(66)
Also the defining relation (63) yields
(120583 + 119901) 119891 (119911) = 120583F (119911) + 119911F1015840(119911) (67)
Now a routine calculation using (65) (66) and (67) gives
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
(119911 isin U)
(68)
Since 119891 isin H119899119898119901(120582 119905 ℎ) we get the following from the
preceding equation (68)
119870 (119911) +
119911119870
1015840(119911)
120583 + 119901
≺ ℎ (119911) (69)
Therefore by applying Lemma 2 we have
119870 (119911) ≺ℎ (119911) = (120583 + 119901) 119911
minus(120583+119901)int
119911
0
119905
120583+119901minus1ℎ (119905) 119889119905 ≺ ℎ (119911)
(70)
This last subordination (70) is equivalent to
F isin H119899119898
119901(120582 119905
ℎ) sub H
119899119898
119901(120582 119905 ℎ) (71)
The proof of Theorem 12 is completed
Theorem 13 Let 119891 isin A119901 and suppose that the function F isdefined as in (63) If
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (120574 gt 0)
(72)
thenF isin H119899119898119901(120582 0
ℎ) where
ℎ (119911) =
120583 + 119901
120574
119911
minus(120583+119901)120574
times int
119911
0
120585
((120583+119901)120574)minus1ℎ (120585) 119889120585 ≺ ℎ (119911) (R (120583) gt minus119901)
(73)
The function ℎ(119911) is the best dominant in (73)
Journal of Complex Analysis 7
Proof We define the function119867 on U by
119867(119911) = 119911
minus119901D(120582119901)
119905 (119899119898)F (119911) (74)
Differentiation of both the sides of (74) combined with theidentity
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
119901
= D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)(75)
gives
119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
) = 119867 (119911) +
119911119867
1015840(119911)
119901
(76)
By making use of (67) we simplify the subordinate of (72) asfollows
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus 120574)119867 (119911) + 120574119911
minus119901D(120582119901)
119905 (119899119898)
times (
120583
120583 + 119901
F (119911) +
119901
120583 + 119901
119911F1015840 (119911)
119901
)
= (1 minus 120574)119867 (119911)
120583120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+
119901120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (
119911F1015840 (119911)
119901
)
(77)
Next by using (74) and (76) the above identity furthersimplifies to the following
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus
119901120574
120583 + 119901
)119867 (119911) +
119901120574
120583 + 119901
(119867 (119911) +
119911119867
1015840(119911)
119901
)
= 119867 (119911) +
120574
120583 + 119901
119911119867
1015840(119911)
(78)
The subordination (72) is thus equivalent to
119867(119911) +
120574
120583 + 119901
119911119867
1015840(119911) ≺ ℎ (119911)
(R (120583) gt minus119901 120574 gt 0 119911 isin U)
(79)
Therefore an application of Lemma 2 yields the assertionof Theorem 13 The proof of Theorem 13 is completed
5 Inclusion Theorems
Theorem 14 Let 119905 gt 0 120574 gt 0 and 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574)
If 120574 le 1205740 then 119891(119911) isin H119899119898119901(120582 0 ℎ) where
1205740 =1
2
(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
minus1
(80)
The bound 1205740 is sharp when ℎ(119911) = 1(1 minus 119911)
Proof Let 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) By setting
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) (119911 isin U) (81)
we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119892 (119911) +
119905
119901
119911119892
1015840(119911)
(82)
Therefore by using Definition 1 we get
119892 (119911) +
119905
119901
119911119892
1015840(119911) ≺ 120574ℎ (119911) + 1 minus 120574 (83)
and an application of Lemma 2 yields
119892 (119911) ≺
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1ℎ (120585) 119889120585 + 1 minus 120574 = (ℎ lowast 120595) (119911)
(84)
where
120595 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574 (85)
It follows from (85) that if 0 lt 120574 le 1205740 where 1205740 gt 1 is givenby (80) then
R (120595 (119911)) =
120574119901
119905
int
1
0
119906
(119901119905)minus1R(
1
1 minus 119906119911
) 119889119906 + 1 minus 120574
gt
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906 + 1 minus 120574
ge
1
2
(119911 isin U)
(86)
Equivalently
120595 (119911) ≺
1
1 minus 119911
(119911 isin U) (87)
Since ℎ(119911) and 120595(119911) are both convex univalent functions inU using Lemma 4 we obtain from (84) that
119892 (119911) ≺ (ℎ lowast 120595) (119911) ≺ ℎ (119911) (88)
Therefore in view of (81) we have
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (89)
Or equivalently 119891(119911) isin H119899119898119901(120582 0 ℎ)
In order to prove that the bound on 1205740 is the best possibleset ℎ(119911) = 1(1 minus 119911) and let the function 119891 be defined onU by
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574
(90)
8 Journal of Complex Analysis
It can be readily verified that 119891 isin A119901 and
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 120574ℎ (119911) + 1 minus 120574
(91)
Thus 119891(119911) isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) Also for 120574 gt 1205740 and
119911 rarr minus1 we have
R 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) 997888rarr
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906
+ 1 minus 120574 lt
1
2
(92)
which shows that
119891 (119911) notin H119899119898
119901(120582 0 ℎ) (93)
Hence the bound 1205740 cannot be increased when ℎ(119911) = 1(1 minus119911) The proof of Theorem 14 is completed
Taking119898 = 0 we use the notation
H1198990
119901(120582 119905 ℎ) = H
119899
119901(120582 119905 ℎ) (94)
in the following theorem
Theorem 15 Let 0 le 1199051 lt 1199052 Then
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (95)
Proof Let 0 le 1199051 lt 1199052 and suppose that 119891 isin H119899119901(120582 1199052 ℎ) We
define
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899 0) 119891 (119911) (119911 isin U) (96)
The function 119892 is analytic in U with 119892(0) = 1 Differenti-ating the expressions on both the sides of (96) with respect to119911 and using Definition 1 we have
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
= 119892 (119911) +
1199052
119901
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U)
(97)
Hence an application of Lemma 2 yields
119892 (119911) ≺ ℎ (119911) (119911 isin U) (98)
Noting that 0 le 11990511199052 lt 1 and ℎ(119911) is convex univalent in Uit follows from (96) (97) and (98) that
(1 minus 1199051) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199051
119911
minus119901+1(D(120582119901)
119905 (119899 0)119891(119911))
1015840
=
1199051
1199052
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
+ (1 minus
1199051
1199052
)119892 (119911)
≺ ℎ (119911) (119911 isin U)
(99)
Thus 119891 isin H119899119901(120582 1199051 ℎ) Consequently
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (100)
The proof of Theorem 15 is completed
Remark 16 The particular case 119899 = 1 of Theorem 15 gives aresult of Liu ([17] Theorem 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Patel and A K Mishra ldquoOn certain subclasses of multivalentfunctions associated with an extended fractional differintegraloperatorrdquo Journal of Mathematical Analysis and Applicationsvol 332 no 1 pp 109ndash122 2007
[2] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987
[3] H M Srivastava and S Owa ldquoAn application of the fractionalderivativerdquo Mathematica Japonica vol 29 no 3 pp 383ndash3891984
[4] A K Mishra and P Gochhayat ldquoApplications of the Owa-Srivastava operator to the class of 119896-uniformly convex func-tionsrdquo Fractional Calculus amp Applied Analysis vol 9 no 4 pp323ndash331 2006
[5] A K Mishra and P Gochhayat ldquoThe Fekete-Szego problem fork-uniformly convex functions and for a class defined by theOwa-Srivastava operatorrdquo Journal ofMathematical Analysis andApplications vol 347 no 2 pp 563ndash572 2008
[6] H M Srivastava A K Mishra and S N Kund ldquoCertainclasses of analytic functions associated with iterations of theOwa-Srivastava fractional derivative operatorrdquo Southeast AsianBulletin of Mathematics vol 37 no 3 pp 413ndash435 2013
[7] HM Srivastava andMK Aouf ldquoA certain fractional derivativeoperator and its applications to a new class of analytic andmultivalent functions with negative coefficientsrdquo Journal ofMathematical Analysis and Applications vol 171 no 1 pp 1ndash131992
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Complex Analysis 7
Proof We define the function119867 on U by
119867(119911) = 119911
minus119901D(120582119901)
119905 (119899119898)F (119911) (74)
Differentiation of both the sides of (74) combined with theidentity
119911(D(120582119901)
119905 (119899119898)F(119911))
1015840
119901
= D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
)(75)
gives
119911
minus119901D(120582119901)
119905 (119899119898)(
119911F1015840 (119911)
119901
) = 119867 (119911) +
119911119867
1015840(119911)
119901
(76)
By making use of (67) we simplify the subordinate of (72) asfollows
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus 120574)119867 (119911) + 120574119911
minus119901D(120582119901)
119905 (119899119898)
times (
120583
120583 + 119901
F (119911) +
119901
120583 + 119901
119911F1015840 (119911)
119901
)
= (1 minus 120574)119867 (119911)
120583120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+
119901120574
120583 + 119901
119911
minus119901D(120582119901)
119905 (
119911F1015840 (119911)
119901
)
(77)
Next by using (74) and (76) the above identity furthersimplifies to the following
(1 minus 120574) 119911
minus119901D(120582119901)
119905 (119899119898)F (119911)
+ 120574119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
= (1 minus
119901120574
120583 + 119901
)119867 (119911) +
119901120574
120583 + 119901
(119867 (119911) +
119911119867
1015840(119911)
119901
)
= 119867 (119911) +
120574
120583 + 119901
119911119867
1015840(119911)
(78)
The subordination (72) is thus equivalent to
119867(119911) +
120574
120583 + 119901
119911119867
1015840(119911) ≺ ℎ (119911)
(R (120583) gt minus119901 120574 gt 0 119911 isin U)
(79)
Therefore an application of Lemma 2 yields the assertionof Theorem 13 The proof of Theorem 13 is completed
5 Inclusion Theorems
Theorem 14 Let 119905 gt 0 120574 gt 0 and 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574)
If 120574 le 1205740 then 119891(119911) isin H119899119898119901(120582 0 ℎ) where
1205740 =1
2
(1 minus
119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906)
minus1
(80)
The bound 1205740 is sharp when ℎ(119911) = 1(1 minus 119911)
Proof Let 119891 isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) By setting
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) (119911 isin U) (81)
we have
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 119892 (119911) +
119905
119901
119911119892
1015840(119911)
(82)
Therefore by using Definition 1 we get
119892 (119911) +
119905
119901
119911119892
1015840(119911) ≺ 120574ℎ (119911) + 1 minus 120574 (83)
and an application of Lemma 2 yields
119892 (119911) ≺
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1ℎ (120585) 119889120585 + 1 minus 120574 = (ℎ lowast 120595) (119911)
(84)
where
120595 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574 (85)
It follows from (85) that if 0 lt 120574 le 1205740 where 1205740 gt 1 is givenby (80) then
R (120595 (119911)) =
120574119901
119905
int
1
0
119906
(119901119905)minus1R(
1
1 minus 119906119911
) 119889119906 + 1 minus 120574
gt
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906 + 1 minus 120574
ge
1
2
(119911 isin U)
(86)
Equivalently
120595 (119911) ≺
1
1 minus 119911
(119911 isin U) (87)
Since ℎ(119911) and 120595(119911) are both convex univalent functions inU using Lemma 4 we obtain from (84) that
119892 (119911) ≺ (ℎ lowast 120595) (119911) ≺ ℎ (119911) (88)
Therefore in view of (81) we have
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) ≺ ℎ (119911) (89)
Or equivalently 119891(119911) isin H119899119898119901(120582 0 ℎ)
In order to prove that the bound on 1205740 is the best possibleset ℎ(119911) = 1(1 minus 119911) and let the function 119891 be defined onU by
119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) =
120574119901
119905
119911
minus119901119905int
119911
0
120585
(119901119905)minus1
1 minus 120585
119889120585 + 1 minus 120574
(90)
8 Journal of Complex Analysis
It can be readily verified that 119891 isin A119901 and
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 120574ℎ (119911) + 1 minus 120574
(91)
Thus 119891(119911) isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) Also for 120574 gt 1205740 and
119911 rarr minus1 we have
R 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) 997888rarr
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906
+ 1 minus 120574 lt
1
2
(92)
which shows that
119891 (119911) notin H119899119898
119901(120582 0 ℎ) (93)
Hence the bound 1205740 cannot be increased when ℎ(119911) = 1(1 minus119911) The proof of Theorem 14 is completed
Taking119898 = 0 we use the notation
H1198990
119901(120582 119905 ℎ) = H
119899
119901(120582 119905 ℎ) (94)
in the following theorem
Theorem 15 Let 0 le 1199051 lt 1199052 Then
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (95)
Proof Let 0 le 1199051 lt 1199052 and suppose that 119891 isin H119899119901(120582 1199052 ℎ) We
define
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899 0) 119891 (119911) (119911 isin U) (96)
The function 119892 is analytic in U with 119892(0) = 1 Differenti-ating the expressions on both the sides of (96) with respect to119911 and using Definition 1 we have
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
= 119892 (119911) +
1199052
119901
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U)
(97)
Hence an application of Lemma 2 yields
119892 (119911) ≺ ℎ (119911) (119911 isin U) (98)
Noting that 0 le 11990511199052 lt 1 and ℎ(119911) is convex univalent in Uit follows from (96) (97) and (98) that
(1 minus 1199051) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199051
119911
minus119901+1(D(120582119901)
119905 (119899 0)119891(119911))
1015840
=
1199051
1199052
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
+ (1 minus
1199051
1199052
)119892 (119911)
≺ ℎ (119911) (119911 isin U)
(99)
Thus 119891 isin H119899119901(120582 1199051 ℎ) Consequently
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (100)
The proof of Theorem 15 is completed
Remark 16 The particular case 119899 = 1 of Theorem 15 gives aresult of Liu ([17] Theorem 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Patel and A K Mishra ldquoOn certain subclasses of multivalentfunctions associated with an extended fractional differintegraloperatorrdquo Journal of Mathematical Analysis and Applicationsvol 332 no 1 pp 109ndash122 2007
[2] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987
[3] H M Srivastava and S Owa ldquoAn application of the fractionalderivativerdquo Mathematica Japonica vol 29 no 3 pp 383ndash3891984
[4] A K Mishra and P Gochhayat ldquoApplications of the Owa-Srivastava operator to the class of 119896-uniformly convex func-tionsrdquo Fractional Calculus amp Applied Analysis vol 9 no 4 pp323ndash331 2006
[5] A K Mishra and P Gochhayat ldquoThe Fekete-Szego problem fork-uniformly convex functions and for a class defined by theOwa-Srivastava operatorrdquo Journal ofMathematical Analysis andApplications vol 347 no 2 pp 563ndash572 2008
[6] H M Srivastava A K Mishra and S N Kund ldquoCertainclasses of analytic functions associated with iterations of theOwa-Srivastava fractional derivative operatorrdquo Southeast AsianBulletin of Mathematics vol 37 no 3 pp 413ndash435 2013
[7] HM Srivastava andMK Aouf ldquoA certain fractional derivativeoperator and its applications to a new class of analytic andmultivalent functions with negative coefficientsrdquo Journal ofMathematical Analysis and Applications vol 171 no 1 pp 1ndash131992
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Complex Analysis
It can be readily verified that 119891 isin A119901 and
(1 minus 119905) 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911)
+
119905
119901
119911
minus119901+1(D(120582119901)
119905 (119899119898) 119891 (119911))
1015840
= 120574ℎ (119911) + 1 minus 120574
(91)
Thus 119891(119911) isin H119899119898119901(120582 119905 120574ℎ + 1 minus 120574) Also for 120574 gt 1205740 and
119911 rarr minus1 we have
R 119911
minus119901D(120582119901)
119905 (119899119898) 119891 (119911) 997888rarr
120574119901
119905
int
1
0
119906
(119901119905)minus1
1 + 119906
119889119906
+ 1 minus 120574 lt
1
2
(92)
which shows that
119891 (119911) notin H119899119898
119901(120582 0 ℎ) (93)
Hence the bound 1205740 cannot be increased when ℎ(119911) = 1(1 minus119911) The proof of Theorem 14 is completed
Taking119898 = 0 we use the notation
H1198990
119901(120582 119905 ℎ) = H
119899
119901(120582 119905 ℎ) (94)
in the following theorem
Theorem 15 Let 0 le 1199051 lt 1199052 Then
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (95)
Proof Let 0 le 1199051 lt 1199052 and suppose that 119891 isin H119899119901(120582 1199052 ℎ) We
define
119892 (119911) = 119911
minus119901D(120582119901)
119905 (119899 0) 119891 (119911) (119911 isin U) (96)
The function 119892 is analytic in U with 119892(0) = 1 Differenti-ating the expressions on both the sides of (96) with respect to119911 and using Definition 1 we have
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
= 119892 (119911) +
1199052
119901
119911119892
1015840(119911) ≺ ℎ (119911) (119911 isin U)
(97)
Hence an application of Lemma 2 yields
119892 (119911) ≺ ℎ (119911) (119911 isin U) (98)
Noting that 0 le 11990511199052 lt 1 and ℎ(119911) is convex univalent in Uit follows from (96) (97) and (98) that
(1 minus 1199051) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199051
119911
minus119901+1(D(120582119901)
119905 (119899 0)119891(119911))
1015840
=
1199051
1199052
(1 minus 1199052) 119911minus119901D(120582119901)
119905 (119899 0) 119891 (119911)
+
119901
1199052
119911
minus119901+1(D(120582119901)
119905 (119899 0) 119891 (119911))
1015840
+ (1 minus
1199051
1199052
)119892 (119911)
≺ ℎ (119911) (119911 isin U)
(99)
Thus 119891 isin H119899119901(120582 1199051 ℎ) Consequently
H119899
119901(120582 1199052 ℎ) sub H
119899
119901(120582 1199051 ℎ) (100)
The proof of Theorem 15 is completed
Remark 16 The particular case 119899 = 1 of Theorem 15 gives aresult of Liu ([17] Theorem 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J Patel and A K Mishra ldquoOn certain subclasses of multivalentfunctions associated with an extended fractional differintegraloperatorrdquo Journal of Mathematical Analysis and Applicationsvol 332 no 1 pp 109ndash122 2007
[2] S Owa and H M Srivastava ldquoUnivalent and starlike gener-alized hypergeometric functionsrdquo Canadian Journal of Mathe-matics vol 39 no 5 pp 1057ndash1077 1987
[3] H M Srivastava and S Owa ldquoAn application of the fractionalderivativerdquo Mathematica Japonica vol 29 no 3 pp 383ndash3891984
[4] A K Mishra and P Gochhayat ldquoApplications of the Owa-Srivastava operator to the class of 119896-uniformly convex func-tionsrdquo Fractional Calculus amp Applied Analysis vol 9 no 4 pp323ndash331 2006
[5] A K Mishra and P Gochhayat ldquoThe Fekete-Szego problem fork-uniformly convex functions and for a class defined by theOwa-Srivastava operatorrdquo Journal ofMathematical Analysis andApplications vol 347 no 2 pp 563ndash572 2008
[6] H M Srivastava A K Mishra and S N Kund ldquoCertainclasses of analytic functions associated with iterations of theOwa-Srivastava fractional derivative operatorrdquo Southeast AsianBulletin of Mathematics vol 37 no 3 pp 413ndash435 2013
[7] HM Srivastava andMK Aouf ldquoA certain fractional derivativeoperator and its applications to a new class of analytic andmultivalent functions with negative coefficientsrdquo Journal ofMathematical Analysis and Applications vol 171 no 1 pp 1ndash131992
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Complex Analysis 9
[8] H M Srivastava and A K Mishra ldquoA fractional differintegraloperator and its applications to a nested class of multivalentfunctions with negative coefficientsrdquo Advanced Studies in Con-temporary Mathematics vol 7 no 2 pp 203ndash214 2003
[9] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
[10] F M Al-Oboudi and K A Al-Amoudi ldquoSubordination resultsfor classes of analytic functions related to conic domains definedby a fractional operatorrdquo Journal of Mathematical Analysis andApplications vol 354 no 2 pp 412ndash420 2009
[11] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis Proceedings of the 5th Romanian Finnish Seminar vol1013 of Lecture Notes in Mathematics pp 362ndash372 SpringerBucharest Romania 1983
[12] P L Duren Univalent Functions vol 259 Springer New YorkNY USA 1983
[13] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
[14] A K Mishra and M M Soren ldquoCertain subclasses of mul-tivalent meromorphic functions involving iterations of theCho-Kwon-Srivastava transform and its combinationsrdquo Asian-European Journal of Mathematics In press
[15] A K Mishra T Panigrahi and R K Mishra ldquoSubordinationand inclusion theorems for subclasses of meromorphic func-tions with applications to electromagnetic cloakingrdquo Mathe-matical and Computer Modelling vol 57 no 3-4 pp 945ndash9622013
[16] J L Liu ldquoSubordinations for certain multivalent analyticfunctions associated with the generalized Srivastava-Attiyaoperatorrdquo Integral Transforms and Special Functions vol 19 no11-12 pp 893ndash901 2008
[17] J-L Liu ldquoOn a subclass of multivalent analytic functionsassociated with extended fractional differintegral operatorrdquoBulletin of the Iranian Mathematical Society In press
[18] J-L Liu and HM Srivastava ldquoA linear operator and associatedfamilies of meromorphically multivalent functionsrdquo Journal ofMathematical Analysis and Applications vol 259 no 2 pp 566ndash581 2001
[19] J-L Liu and H M Srivastava ldquoA class of multivalently analyticfunctions associated with the Dziok-Srivastava operatorrdquo Inte-gral Transforms and Special Functions vol 20 no 5-6 pp 401ndash417 2009
[20] H M Srivastava M Darus and R W Ibrahim ldquoClasses ofanalytic functions with fractional powers defined by meansof a certain linear operatorrdquo Integral Transforms and SpecialFunctions vol 22 no 1 pp 17ndash28 2011
[21] R Singh and S Singh ldquoConvolution properties of a class ofstarlike functionsrdquo Proceedings of the American MathematicalSociety vol 106 no 1 pp 145ndash152 1989
[22] D G Yang ldquoProperties of a class of analytic functionsrdquoMathematica Japonica vol 41 no 2 pp 371ndash381 1995
[23] Z Zhongzhu and S Owa ldquoConvolution properties of a class ofbounded analytic functionsrdquo Bulletin of the Australian Mathe-matical Society vol 45 no 1 pp 9ndash23 1992
[24] L Jinlin ldquoOn subordination for certain subclass of analyticfunctionsrdquo International Journal ofMathematics andMathemat-ical Sciences vol 20 no 2 pp 225ndash228 1997
[25] S RuscheweyhConvolutions in Geometric FunctionTheory LesPresses de IrsquoUniversite de Montreal 1982
[26] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash171 1981
[27] D J Hallenbeck and S Ruscheweyh ldquoSubordination by convexfunctionsrdquo Proceedings of the American Mathematical Societyvol 52 pp 191ndash195 1975
[28] S Ruscheweyh and J Stankiewicz ldquoSubordination under con-vex univalent functionsrdquo Bulletin of the Polish Academy ofSciences Mathematics vol 33 no 9-10 pp 499ndash502 1985
[29] T H MacGregor ldquoFunctions whose derivative has a positivereal partrdquo Transactions of the American Mathematical Societyvol 104 pp 532ndash537 1962
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
