Research Article On Certain Subclasses of Analytic Multivalent...
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Research Article On Certain Subclasses of Analytic Multivalent Functions Using Generalized Salagean Operator Adnan Ghazy Alamoush and Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600 Bangi, Selangor, Malaysia Correspondence should be addressed to Maslina Darus; [email protected] Received 16 February 2015; Revised 24 May 2015; Accepted 21 June 2015 Academic Editor: Patricia J. Y. Wong Copyright © 2015 A. G. Alamoush and M. Darus. 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. We introduce and study two subclasses of multivalent functions denoted by M ,,, , ( 1 ; 2 ) and N ,,, , (, ; ). Further, by using the method of differential subordination, certain inclusion relations between the two subclasses aforementioned are given. Moreover, several consequences of the main results are also discussed. 1. Introduction Let A (,) denote the class of the functions of the form () = + ∞ ∑ =+ , (, ∈ N ={1, 2, 3, . . .}) , (1) which are analytic in the open unit disc U = { ∈ C : || < 1}, and let denote A := A (1,1) . A function ∈ A (,) is said to be multivalent starlike functions of order in U, if it satisfies the following inequal- ity: R { () () } > , ∈ U,(0 ≤ < , ∈ N), (2) and we denote this class by ∗ , (). A function ∈ A (,) is said to be multivalent convex functions of order in U, if it satisfies the following inequal- ity: R {1 + () () } > , ∈ U,(0 ≤ < , ∈ N), (3) and we denote this class by , (). For a function ∈ A (,) , Goyal et al. [1] introduced the following generalized Salagean differential operator: 0 () = () , (4) 1 () = () = (1 − ) () + () , ( ≥ 0), (5) () = ( −1 ()), ( ∈ N). (6) If is given by (1), then from (5) and (6) we have () = + ∞ ∑ =+ [1 +( − 1) ] . (7) Remark 1. For == 1, the differential operator () reduces to Salagean differential operator () [2]. Definition 2. Let M ,,, , ( 1 ; 2 ) be the class of functions ∈ A (,) that satisfy the condition Hindawi Publishing Corporation International Journal of Differential Equations Volume 2015, Article ID 910124, 7 pages http://dx.doi.org/10.1155/2015/910124
Transcript of Research Article On Certain Subclasses of Analytic Multivalent...
Research ArticleOn Certain Subclasses of Analytic Multivalent Functions UsingGeneralized Salagean Operator
Adnan Ghazy Alamoush and Maslina Darus
School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia (UKM)43600 Bangi Selangor Malaysia
Correspondence should be addressed to Maslina Darus maslinaukmedumy
Received 16 February 2015 Revised 24 May 2015 Accepted 21 June 2015
Academic Editor Patricia J Y Wong
Copyright copy 2015 A G Alamoush and M Darus This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We introduce and study two subclasses of multivalent functions denoted by M119898120572120573120590119901119899
(1205821 1205822) and N119898120572120573120590
119901119899(120583 120575 120574) Further by
using the method of differential subordination certain inclusion relations between the two subclasses aforementioned are givenMoreover several consequences of the main results are also discussed
1 Introduction
LetA(119901119899)
denote the class of the functions 119891 of the form
119891 (119911) = 119911119901
+
infin
sum119896=119901+119899
119886119896119911119896
(119899 119901 isin N = 1 2 3 ) (1)
which are analytic in the open unit discU = 119911 isin C |119911| lt 1and let denoteA = A
(11)A function 119891 isin A
(119901119899)is said to be multivalent starlike
functions of order 120572 in U if it satisfies the following inequal-ity
R1199111198911015840
(119911)
119891 (119911) gt 120572 119911 isin U (0 le 120572 lt 119901 119901 isin N) (2)
and we denote this class by 119878lowast119901119899(120572)
A function 119891 isin A(119901119899)
is said to be multivalent convexfunctions of order 120572 in U if it satisfies the following inequal-ity
R1+11991111989110158401015840 (119911)
1198911015840 (119911) gt 120572 119911 isin U (0 le 120572 lt 119901 119901 isin N) (3)
and we denote this class by 119862119901119899(120572)
For a function 119891 isin A(119901119899)
Goyal et al [1] introduced thefollowing generalized Salagean differential operator
1198630120590119891 (119911) = 119891 (119911) (4)
1198631120590119891 (119911) = 119863
120590119891 (119911) = (1minus120590) 119891 (119911) + 120590
1199011199111198911015840
(119911)
(120590 ge 0) (5)
119863119898
120590119891 (119911) = 119863
120590(119863119898minus1120590119891 (119911)) (119898 isin N) (6)
If 119891 is given by (1) then from (5) and (6) we have
119863119898
120590119891 (119911) = 119911
119901
+
infin
sum119896=119901+119899
[1+(119896119901minus 1)120590]
119898
119886119896119911119896
(7)
Remark 1 For 120590 = 119901 = 1 the differential operator 119863119898120590119891(119911)
reduces to Salagean differential operator119863119898119891(119911) [2]
Definition 2 Let M119898120572120573120590119901119899
(1205821 1205822) be the class of functions119891 isin A
(119901119899)that satisfy the condition
Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2015 Article ID 910124 7 pageshttpdxdoiorg1011552015910124
2 International Journal of Differential Equations
R
(1minus1205821)119911 (119863120572120573119898120590
119891 (119911))1015840
119863120572120573119898
120590119891 (119911)
+ 1205821(1+119911 (119863120572120573119898120590
119891 (119911))10158401015840
(119863120572120573119898
120590119891 (119911))
1015840)
gt 1205822 119911 isin U
(8)
where (0 le 120590 0 le 120572 lt 120573 le 1 1205821 isin R 0 le 1205822 lt 119901 119898 119901 isinN) and let N119898120572120573120590
119901119899(120583 120575 120574) be the class of functions 119891 isin
A(119901119899)
that satisfy the conditions
(119863120572120573119898120590
119891 (119911)) (119863120572120573119898
120590119891 (119911))
1015840
1199112119901minus1= 0 119911 isin U
R
(119863120572120573119898
120590119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
gt 120574 119911 isin U
(0 le 120590 120575 120583 isin R 0 le 120572 lt 120573 le 1 0 le 120574 lt 119901120575 (120572 + 120573119901)120575+120583 119898 119901 isin N)
(9)
where
119863120572120573119898
120590119891 (119911) = 120572119863
119898
120590119891 (119911) + 120573119911 (119863
119898
120590119891 (119911))
1015840
(10)
Remark 3 By specifying different values we have some well-known subclasses of the classes 119860
(119901119899)and 119860
(119899)= 119860(1119899)
appearing from the families of the classes M119898120572120573120590119901119899
(1205821 1205822)
andN119898120572120573120590
119901119899(120583 120575 120574)
(i) M010120590119901119899
(0 1205821) = N010120590119901119899
(minus1 1 1205821) = 119878lowast
119901119899 (0 le 1205821 lt
119901) is the class ofmultivalent starlike functions of order1205821
(ii) M0101205901119899 (0 1205821) = N010120590
1119899 (minus1 1 1205821) = 119878lowast
1119899 = 119878lowast
119899 (0 le
1205821 lt 1) is the class of starlike functions of order 1205821(iii) M1011
119901119899(0 1205821) = C
119901119899(1205821) (0 le 1205821 lt 119901) is the class
of multivalent convex functions of order 1205821(iv) M1011
1119899 (0 1205821) = C1119899(1205821) = C119899(1205821) (0 le 1205821 lt 1) is
the class of convex functions of order 120574(v) N1101
1119899 (1 120575 1205821) =B119899(120575 1205821) (120575 ge minus1 0 le 1205821 lt 1) is
the subclass of Bazilevic functions
LetH[119886 119899] be denoted by the class
H [119886 119899]
= ℎ isinH (U) ℎ (119911) = 119886 + 119886119899119911119899
+ sdot sdot sdot 119911 isinU (11)
In this investigation we focus on certain inequalities con-sisting of the following differential operator J119898120572120573120590
119901119899(120583 120575)
A(119901119899)
rarr H[(120583 + 120575) 119899 + 119901]
J119898120572120573120590
119901119899(120583 120575) 119891 (119911) = 120583
119911 (119863120572120573119898120590
119891 (119911))1015840
119863120572120573119898
120590119891 (119911)
+ 120575(1+119911 (119863120572120573119898120590
119891 (119911))10158401015840
(119863120572120573119898
120590119891 (119911))
1015840)
(12)
that generalizes the expression used in the definition of classM119898120572120573120590
119901119899(1205821 1205822) and we receive several properties of the
expression
(119863120572120573119898120590
119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
(119911 isin U) (13)
including relations between classes M119898120572120573120590
119901119899(1205821 1205822) and
N119898120572120573120590
119901119899(120583 120575 120574)
In order to prove our main results we will need thefollowing lemmas due to Miller and Mocanu [3]
Lemma 4 Let Ω sub C and suppose that the function 120595 C2 times
U rarr C satisfies 120595(119872119890119894120579 119870119890119894120579 119911) notin Ω for all 119870 ge 119872119899 120579 isinR and 119911 isin U If ℎ(119911) = 119886 + ℎ
119899119911119899 + sdot sdot sdot is analytic in U and
120595(ℎ(119911) 119911ℎ1015840(119911) 119911) isin Ω for all 119911 isin U then |ℎ(119911)| lt 119872 119911 isin U
Lemma 5 Let Ω sub C and suppose that the function 120595 C2 timesU rarr C satisfies 120595(119894119909 119910 119911) notin Ω for all 119909 isin R 119910 le minus119899(1 +1199092)2 and 119911 isin U If ℎ(119911) = 119886 + ℎ
119899119911119899 + sdot sdot sdot is analytic in U and
120595(ℎ(119911) 119911ℎ1015840(119911) 119911) isin Ω for all 119911 isin U thenRℎ(119911) gt 0 119911 isin U
2 Main Results
Following the same techniques and procedure given byGoswami et al [4] we have the following results
Theorem6 Let119891(119911) isin A(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 120583 120575 isin R If
R J119898120572120573120590
119901119899(120583 120575) 119891 (119911)
lt 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (120572 + 120573119901)120575+120583
(119911 isin U) (14)
International Journal of Differential Equations 3
where 119901120575(120572 + 120573119901)120575+120583 le 119872 then
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863120572120573119898120590
119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus119901120575
(120572 + 120573119901)120575+120583
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872 (119911 isin U)
(15)
where the powers are the principal ones
Proof Let the function ℎ(119911) be defined by
ℎ (119911) = (119863120572120573119898120590
119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus119901120575
(120572 +120573119901)120575+120583
(16)
From the assumptions 119891 isin A(119901119899)
with(119863120572120573119898120590
119891(119911))(119863120572120573119898120590
119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U wehave that ℎ isinH[0 119899] By a simple manipulation we have
J119898120572120573120590
119901119899(120583 120575) 119891 (119911) = 119901 (120575 + 120583)
+119911ℎ1015840 (119911)
ℎ (119911) + 119901120575 (120572 + 120573119901)120575+120583
(17)
Now letting
120595 (119903 119904 119911) = 119901 (120575 + 120583) +119904
119903 + 119901120575 (120572 + 120573119901)120575+120583
Ω = 119908isinC R (119908) lt 119901 (120575 + 120583)
+119899119872
119872 + 119901120575 (120572 + 120573119901)120575+120583
(18)
we have from (17) and (14) that
120595 (ℎ (119911) 119911ℎ1015840
(119911) 119911) = J119898120572120573120590
119901119899(120583 120575) 119891 (119911) isin Ω
forall119911 isin U
(19)
Further for any 120579 isin R 119870 ge 119899119872 and 119911 isin U since 119872 ge
119901120575(120572 + 120573119901)120575+120583 we also have
R 120595 (119872119890119894120579
119870119890119894120579
119911)
= 119901 (120575 + 120583) +119870R(1
119872+ 119890minus119894120579119901120575 (120572 + 120573119901)120575+120583
)
ge 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (120572 + 120573119901)120575+120583
(119911 isin U)
(20)
which shows that 120595(119872119890119894120579 119870119890119894120579 119911) notin Ω for all 120579 isin R 119870 ge119899119872 and 119911 isin U Therefore according to Lemma 4 we obtain|ℎ(119911)| lt 119872 (119911 isin U) Hence (15) is proven
Theorem7 Let119891(119911) isin A(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 120583 120575 isin R If
R J119898120572120573120590
119901119899(120583 120575) 119891 (119911) gt 119896 (120583 120575 120572 120573 120574)
(119911 isin U)
(21)
where 120574 isin [0 119901120575(120572 + 120573119901)120575+120583) and
119896 (120583 120575 120572 120573 120574) =
119901 (120575 + 120583) minus119899120574
2 [119901120575 (120572 + 120573119901)120575+120583 minus 120574] if 120574 isin [0
119901120575 (120572 + 120573119901)120575+120583
2]
119901 (120575 + 120583) minus119899 [119901120575 (120572 + 120573119901)
120575+120583
minus 120574]
2120574 if 120574 isin [
119901120575 (120572 + 120573119901)
120575+120583
2 119901120575 (120572 + 120573119901)
120575+120583
)
(22)
then 119891 isinN119898120590120572120573119901119899
(120583 120575 120574)
Proof Suppose that
ℎ (119911) =1
119901120575 (120572 + 120573119901)120575+120583
minus 120574
[[
[
(119863120572120573119898
120590119891 (119911)
119911119901)
120583
sdot ((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus 120574]]
]
(23)
Then ℎ(119911) = 1 + ℎ119899119911119899 + sdot sdot sdot is analytic in U It is easily seen
from (23) that
4 International Journal of Differential Equations
J119898120572120573120590
119901119899(120583 120575) 119891 (119911)
= 119901 (120575 + 120583) +(119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119911ℎ1015840 (119911)
(119901120575 (120572 + 120573119901)120575+120583
minus 120574) ℎ (119911) + 120574
(24)
Further since
120595 (119903 119904 119911) = 119901 (120575 + 120583) +(119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119904
(119901120575 (120572 + 120573119901)120575+120583
minus 120574) 119903 + 120574
Ω = 119908isinC R (119908) gt 119896 (120583 120575 120572 120573 120574)
(25)
it leads to
120595 (ℎ (119911) 119911ℎ1015840
(119911) 119911) = J119898120572120573120590
119901119899(120583 120575) 119891 (119911) isin Ω
forall119911 isin U
(26)
Also for any 119909 isin R 119910 le minus119899(1 + 1199092)2 and 119911 isin U we have
R 120595 (119894119909 119910 119911) = 119901 (120575 + 120583) +120574 (119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119910
[119901120575 (120572 + 120573119901)120575+120583
minus 120574]21199092 + 1205742
le 119901 (120575 + 120583)
minus119899120574 [119901120575 (120572 + 120573119901)
120575+120583
minus 120574]
21 + 1199092
[119901120575 (120572 + 120573119901)120575+120583
minus 120574]21199092 + 1205742
equiv 119902 (119911) le 119896 (120583 120575 120572 120573 120574)
=
lim119909rarrinfin
119902 (119911) if 120574 isin [0119901120575 (120572 + 120573119901)
120575+120583
2]
119902 (0) if 120574 isin [119901120575 (120572 + 120573119901)
120575+120583
2 119901120575 (120572 + 120573119901)
120575+120583
)
(27)
that is 120595(119894119909 119910 119911) notin Ω Finally by Lemma 5 we obtain that119877119890(ℎ(119911)) gt 0 The proof of Theorem 7 is complete
3 Corollaries and Consequences
We will discuss some interesting consequences of the maintheorems that extend some previous results obtained in ([45])
Putting 120572 = 1 120573 = 0 in Theorems 6 and 7 we get thefollowing corollaries
Corollary 8 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also
let 120583 120575 isin R If
R120583119911 (119863119898120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 120575(1+119911 (119863119898
120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840)
lt 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (119911 isin U)
(28)
where 119901120575 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863119898120590119891 (119911)
119911119901)
120583
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120575
minus119901120575
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(29)
where the powers are the principal ones
Corollary 9 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also
let 120583 120575 isin R If
R120583119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 120575(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840)
gt 120593 (120583 120575 120574) (119911 isin U)
(30)
where 120574 isin [0 119901120575) and
120593 (120583 120575 120574) = 119896 (120583 120575 1 0 120574)
=
119901 (120575 + 120583) minus119899120574
2 [119901120575 minus 120574] 119894119891 120574 isin [0
119901120575
2]
119901 (120575 + 120583) minus119899 [119901120575 minus 120574]
2120574 119894119891 120574 isin [
119901120575
2 119901120575)
(31)
then
R
(119863119898
120590119891 (119911)
119911119901)
120583
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120575
gt 120574 (119911 isin U) (32)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 8 and 9respectively we obtain the following special cases
Corollary 10 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898
120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) lt 119901+
119899119872
119872 + 1199011205821
(119911 isin U)
(33)
where 1199011205821 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863119898120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
minus1199011205821
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(34)
where the powers are the principal ones
International Journal of Differential Equations 5
Corollary 11 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) gt 120594 (1205821 120574)
(119911 isin U)
(35)
where 120574 isin [0 1199011205821) and
120594 (1205821 120574) = 119896 (1minus1205821 1205821 1 0 120574)
=
119901 minus119899120574
2 [1199011205821 minus 120574] 119894119891 120574 isin [0
1199011205821
2]
119901 minus119899 [1199011205821 minus 120574]
2120574 119894119891 120574 isin [
11990112058212 1199011205821)
(36)
then
R
(119863119898
120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(37)
where the powers are the principal ones
Next upon taking 120572 = 0 120573 = 1 in Theorems 6 and 7 weobtain the following results
Corollary 12 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) lt 119901 (120575 + 120583) +119899119872
119872 + 1199012120575+120583
(119911 isin U)
(38)
where 1199012120575+120583 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120583
((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
minus1199012120575+120583100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872 (119911 isin U)
(39)
where the powers are the principal ones
Corollary 13 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) gt 120601 (120583 120575 120574) (119911 isin U) (40)
where 120574 isin [0 1199012120575+120583) and
120601 (120583 120575 120574) = 119896 (120583 120575 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199012120575+120583 minus 120574] 119894119891 120574 isin [0
1199012120575+120583
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199012120575+120583
2 1199012120575+120583)
(41)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
120583
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
gt 120574
(119911 isin U)
(42)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 12 and 13respectively we obtain the following special cases
Corollary 14 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) lt 119901+119899119872
119872 + 1199011205821+1
(119911 isin U)
(43)
where 1199011205821+1 le 119872 then
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
minus1199011205821+1100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(44)
where the powers are the principal ones
Corollary 15 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) gt 120595 (1205821 120574)
(119911 isin U) (45)
where 120574 isin [0 1199011205821+1) and
6 International Journal of Differential Equations
120595 (1205821 120574) = 119896 (1minus1205821 1205821 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199011205821+1 minus 120574] 119894119891 120574 isin [0
1199011205821+1
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199011205821+1
2 1199011205821+1)
(46)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(47)
where the powers are the principal ones
In the next result we will find the relation betweenM119898120572120573120590
119901119899(1205821 120574) andM
119898120572120573120590
119901119899(1 minus 1205821 1205821 120574) For this purpose
taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Theorem 7 we obtain thefollowing result
Corollary 16 Let119891(119911)isinA(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901119899(1205821 984858 (1205821 120572 120573 120574)) (48)
where 120574 isin [0 1199011205821(120572 + 120573119901)) and
984858 (1205821 120572 120573 120574) = 119896 (1minus1205821 1205821 120572 120573 120574) =
119901 minus119899120574
2 [1199011205821 (120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
1199011205821 (120572 + 120573119901)
2]
119901 minus119899 [1199011205821 (120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
1199011205821 (120572 + 120573119901)
2 1199011205821 [(120572 + 120573119901))
(49)
then 119891(119911) isinN119898120572120573120590119901119899
(1 minus 1205821 1205821 120574)
Taking 1205821 = 0 and 119899 = 1 in the above corollary we get thenext special result
Corollary 17 Let119891(119911) isinA(119901)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901(984858 (120572 120573 120574)) (50)
where 120574 isin [0 120572 + 120573119901) and
984858 (120572 120573 120574) = 119896 (1 0 120572 120573 120574)
=
119901 minus120574
2 [(120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
120572 + 120573119901
2]
119901 minus[(120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
120572 + 120573119901
2 (120572 + 120573119901))
(51)
then 119891(119911) isinN119898120572120573120590119901
(1 0 120574)
Again for the special cases of 120583 and 120575 Theorems 6 and 7reduce at once to some results obtained by [4 5]
Remark 18 Taking 119901 = 1 and 119898 = 0 in (7) and 120572 = 1 and120573 = 0 in (10) we get a known result obtained by Irmak et al[5]
Remark 19 Taking119898 = 0 in (7) and 120572 = 1 minus 120573 in (10) we geta known result obtained by Goswami et al [4]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors agreed with the contents of the paper
Acknowledgments
The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia (UKM) under Grant ERGS12013STG06UKM012Thefirst author is also supported byZAMALAHschemegrant under postgraduate center UKM
References
[1] S P Goyal O Singh and P Goswami ldquoSome relations betweencertain classes of analytic multivalent functions involving gen-eralized salagean operatorrdquo Sohag Journal of Mathematics vol1 no 1 pp 27ndash32 2014
[2] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysismdashFifth Romanian-Finnish Seminar Part 1 Proceedingsof the Seminar Held in Bucharest 1981 vol 1013 of Lecture Notesin Mathematics pp 362ndash372 Springer Berlin Germany 1983
[3] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000
International Journal of Differential Equations 7
[4] P Goswami T Bulboaca and S Bansal ldquoSome relationsbetween certain classes of analytic functionsrdquo Journal of Clas-sical Analysis vol 1 no 2 pp 157ndash173 2012
[5] H Irmak T Bulboaca andN Tuneski ldquoSome relations betweencertain classes consisting of 120572-convex type and Bazilevic typefunctionsrdquoAppliedMathematics Letters vol 24 no 12 pp 2010ndash2014 2011
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Differential Equations
R
(1minus1205821)119911 (119863120572120573119898120590
119891 (119911))1015840
119863120572120573119898
120590119891 (119911)
+ 1205821(1+119911 (119863120572120573119898120590
119891 (119911))10158401015840
(119863120572120573119898
120590119891 (119911))
1015840)
gt 1205822 119911 isin U
(8)
where (0 le 120590 0 le 120572 lt 120573 le 1 1205821 isin R 0 le 1205822 lt 119901 119898 119901 isinN) and let N119898120572120573120590
119901119899(120583 120575 120574) be the class of functions 119891 isin
A(119901119899)
that satisfy the conditions
(119863120572120573119898120590
119891 (119911)) (119863120572120573119898
120590119891 (119911))
1015840
1199112119901minus1= 0 119911 isin U
R
(119863120572120573119898
120590119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
gt 120574 119911 isin U
(0 le 120590 120575 120583 isin R 0 le 120572 lt 120573 le 1 0 le 120574 lt 119901120575 (120572 + 120573119901)120575+120583 119898 119901 isin N)
(9)
where
119863120572120573119898
120590119891 (119911) = 120572119863
119898
120590119891 (119911) + 120573119911 (119863
119898
120590119891 (119911))
1015840
(10)
Remark 3 By specifying different values we have some well-known subclasses of the classes 119860
(119901119899)and 119860
(119899)= 119860(1119899)
appearing from the families of the classes M119898120572120573120590119901119899
(1205821 1205822)
andN119898120572120573120590
119901119899(120583 120575 120574)
(i) M010120590119901119899
(0 1205821) = N010120590119901119899
(minus1 1 1205821) = 119878lowast
119901119899 (0 le 1205821 lt
119901) is the class ofmultivalent starlike functions of order1205821
(ii) M0101205901119899 (0 1205821) = N010120590
1119899 (minus1 1 1205821) = 119878lowast
1119899 = 119878lowast
119899 (0 le
1205821 lt 1) is the class of starlike functions of order 1205821(iii) M1011
119901119899(0 1205821) = C
119901119899(1205821) (0 le 1205821 lt 119901) is the class
of multivalent convex functions of order 1205821(iv) M1011
1119899 (0 1205821) = C1119899(1205821) = C119899(1205821) (0 le 1205821 lt 1) is
the class of convex functions of order 120574(v) N1101
1119899 (1 120575 1205821) =B119899(120575 1205821) (120575 ge minus1 0 le 1205821 lt 1) is
the subclass of Bazilevic functions
LetH[119886 119899] be denoted by the class
H [119886 119899]
= ℎ isinH (U) ℎ (119911) = 119886 + 119886119899119911119899
+ sdot sdot sdot 119911 isinU (11)
In this investigation we focus on certain inequalities con-sisting of the following differential operator J119898120572120573120590
119901119899(120583 120575)
A(119901119899)
rarr H[(120583 + 120575) 119899 + 119901]
J119898120572120573120590
119901119899(120583 120575) 119891 (119911) = 120583
119911 (119863120572120573119898120590
119891 (119911))1015840
119863120572120573119898
120590119891 (119911)
+ 120575(1+119911 (119863120572120573119898120590
119891 (119911))10158401015840
(119863120572120573119898
120590119891 (119911))
1015840)
(12)
that generalizes the expression used in the definition of classM119898120572120573120590
119901119899(1205821 1205822) and we receive several properties of the
expression
(119863120572120573119898120590
119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
(119911 isin U) (13)
including relations between classes M119898120572120573120590
119901119899(1205821 1205822) and
N119898120572120573120590
119901119899(120583 120575 120574)
In order to prove our main results we will need thefollowing lemmas due to Miller and Mocanu [3]
Lemma 4 Let Ω sub C and suppose that the function 120595 C2 times
U rarr C satisfies 120595(119872119890119894120579 119870119890119894120579 119911) notin Ω for all 119870 ge 119872119899 120579 isinR and 119911 isin U If ℎ(119911) = 119886 + ℎ
119899119911119899 + sdot sdot sdot is analytic in U and
120595(ℎ(119911) 119911ℎ1015840(119911) 119911) isin Ω for all 119911 isin U then |ℎ(119911)| lt 119872 119911 isin U
Lemma 5 Let Ω sub C and suppose that the function 120595 C2 timesU rarr C satisfies 120595(119894119909 119910 119911) notin Ω for all 119909 isin R 119910 le minus119899(1 +1199092)2 and 119911 isin U If ℎ(119911) = 119886 + ℎ
119899119911119899 + sdot sdot sdot is analytic in U and
120595(ℎ(119911) 119911ℎ1015840(119911) 119911) isin Ω for all 119911 isin U thenRℎ(119911) gt 0 119911 isin U
2 Main Results
Following the same techniques and procedure given byGoswami et al [4] we have the following results
Theorem6 Let119891(119911) isin A(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 120583 120575 isin R If
R J119898120572120573120590
119901119899(120583 120575) 119891 (119911)
lt 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (120572 + 120573119901)120575+120583
(119911 isin U) (14)
International Journal of Differential Equations 3
where 119901120575(120572 + 120573119901)120575+120583 le 119872 then
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863120572120573119898120590
119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus119901120575
(120572 + 120573119901)120575+120583
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872 (119911 isin U)
(15)
where the powers are the principal ones
Proof Let the function ℎ(119911) be defined by
ℎ (119911) = (119863120572120573119898120590
119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus119901120575
(120572 +120573119901)120575+120583
(16)
From the assumptions 119891 isin A(119901119899)
with(119863120572120573119898120590
119891(119911))(119863120572120573119898120590
119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U wehave that ℎ isinH[0 119899] By a simple manipulation we have
J119898120572120573120590
119901119899(120583 120575) 119891 (119911) = 119901 (120575 + 120583)
+119911ℎ1015840 (119911)
ℎ (119911) + 119901120575 (120572 + 120573119901)120575+120583
(17)
Now letting
120595 (119903 119904 119911) = 119901 (120575 + 120583) +119904
119903 + 119901120575 (120572 + 120573119901)120575+120583
Ω = 119908isinC R (119908) lt 119901 (120575 + 120583)
+119899119872
119872 + 119901120575 (120572 + 120573119901)120575+120583
(18)
we have from (17) and (14) that
120595 (ℎ (119911) 119911ℎ1015840
(119911) 119911) = J119898120572120573120590
119901119899(120583 120575) 119891 (119911) isin Ω
forall119911 isin U
(19)
Further for any 120579 isin R 119870 ge 119899119872 and 119911 isin U since 119872 ge
119901120575(120572 + 120573119901)120575+120583 we also have
R 120595 (119872119890119894120579
119870119890119894120579
119911)
= 119901 (120575 + 120583) +119870R(1
119872+ 119890minus119894120579119901120575 (120572 + 120573119901)120575+120583
)
ge 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (120572 + 120573119901)120575+120583
(119911 isin U)
(20)
which shows that 120595(119872119890119894120579 119870119890119894120579 119911) notin Ω for all 120579 isin R 119870 ge119899119872 and 119911 isin U Therefore according to Lemma 4 we obtain|ℎ(119911)| lt 119872 (119911 isin U) Hence (15) is proven
Theorem7 Let119891(119911) isin A(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 120583 120575 isin R If
R J119898120572120573120590
119901119899(120583 120575) 119891 (119911) gt 119896 (120583 120575 120572 120573 120574)
(119911 isin U)
(21)
where 120574 isin [0 119901120575(120572 + 120573119901)120575+120583) and
119896 (120583 120575 120572 120573 120574) =
119901 (120575 + 120583) minus119899120574
2 [119901120575 (120572 + 120573119901)120575+120583 minus 120574] if 120574 isin [0
119901120575 (120572 + 120573119901)120575+120583
2]
119901 (120575 + 120583) minus119899 [119901120575 (120572 + 120573119901)
120575+120583
minus 120574]
2120574 if 120574 isin [
119901120575 (120572 + 120573119901)
120575+120583
2 119901120575 (120572 + 120573119901)
120575+120583
)
(22)
then 119891 isinN119898120590120572120573119901119899
(120583 120575 120574)
Proof Suppose that
ℎ (119911) =1
119901120575 (120572 + 120573119901)120575+120583
minus 120574
[[
[
(119863120572120573119898
120590119891 (119911)
119911119901)
120583
sdot ((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus 120574]]
]
(23)
Then ℎ(119911) = 1 + ℎ119899119911119899 + sdot sdot sdot is analytic in U It is easily seen
from (23) that
4 International Journal of Differential Equations
J119898120572120573120590
119901119899(120583 120575) 119891 (119911)
= 119901 (120575 + 120583) +(119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119911ℎ1015840 (119911)
(119901120575 (120572 + 120573119901)120575+120583
minus 120574) ℎ (119911) + 120574
(24)
Further since
120595 (119903 119904 119911) = 119901 (120575 + 120583) +(119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119904
(119901120575 (120572 + 120573119901)120575+120583
minus 120574) 119903 + 120574
Ω = 119908isinC R (119908) gt 119896 (120583 120575 120572 120573 120574)
(25)
it leads to
120595 (ℎ (119911) 119911ℎ1015840
(119911) 119911) = J119898120572120573120590
119901119899(120583 120575) 119891 (119911) isin Ω
forall119911 isin U
(26)
Also for any 119909 isin R 119910 le minus119899(1 + 1199092)2 and 119911 isin U we have
R 120595 (119894119909 119910 119911) = 119901 (120575 + 120583) +120574 (119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119910
[119901120575 (120572 + 120573119901)120575+120583
minus 120574]21199092 + 1205742
le 119901 (120575 + 120583)
minus119899120574 [119901120575 (120572 + 120573119901)
120575+120583
minus 120574]
21 + 1199092
[119901120575 (120572 + 120573119901)120575+120583
minus 120574]21199092 + 1205742
equiv 119902 (119911) le 119896 (120583 120575 120572 120573 120574)
=
lim119909rarrinfin
119902 (119911) if 120574 isin [0119901120575 (120572 + 120573119901)
120575+120583
2]
119902 (0) if 120574 isin [119901120575 (120572 + 120573119901)
120575+120583
2 119901120575 (120572 + 120573119901)
120575+120583
)
(27)
that is 120595(119894119909 119910 119911) notin Ω Finally by Lemma 5 we obtain that119877119890(ℎ(119911)) gt 0 The proof of Theorem 7 is complete
3 Corollaries and Consequences
We will discuss some interesting consequences of the maintheorems that extend some previous results obtained in ([45])
Putting 120572 = 1 120573 = 0 in Theorems 6 and 7 we get thefollowing corollaries
Corollary 8 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also
let 120583 120575 isin R If
R120583119911 (119863119898120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 120575(1+119911 (119863119898
120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840)
lt 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (119911 isin U)
(28)
where 119901120575 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863119898120590119891 (119911)
119911119901)
120583
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120575
minus119901120575
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(29)
where the powers are the principal ones
Corollary 9 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also
let 120583 120575 isin R If
R120583119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 120575(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840)
gt 120593 (120583 120575 120574) (119911 isin U)
(30)
where 120574 isin [0 119901120575) and
120593 (120583 120575 120574) = 119896 (120583 120575 1 0 120574)
=
119901 (120575 + 120583) minus119899120574
2 [119901120575 minus 120574] 119894119891 120574 isin [0
119901120575
2]
119901 (120575 + 120583) minus119899 [119901120575 minus 120574]
2120574 119894119891 120574 isin [
119901120575
2 119901120575)
(31)
then
R
(119863119898
120590119891 (119911)
119911119901)
120583
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120575
gt 120574 (119911 isin U) (32)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 8 and 9respectively we obtain the following special cases
Corollary 10 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898
120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) lt 119901+
119899119872
119872 + 1199011205821
(119911 isin U)
(33)
where 1199011205821 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863119898120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
minus1199011205821
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(34)
where the powers are the principal ones
International Journal of Differential Equations 5
Corollary 11 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) gt 120594 (1205821 120574)
(119911 isin U)
(35)
where 120574 isin [0 1199011205821) and
120594 (1205821 120574) = 119896 (1minus1205821 1205821 1 0 120574)
=
119901 minus119899120574
2 [1199011205821 minus 120574] 119894119891 120574 isin [0
1199011205821
2]
119901 minus119899 [1199011205821 minus 120574]
2120574 119894119891 120574 isin [
11990112058212 1199011205821)
(36)
then
R
(119863119898
120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(37)
where the powers are the principal ones
Next upon taking 120572 = 0 120573 = 1 in Theorems 6 and 7 weobtain the following results
Corollary 12 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) lt 119901 (120575 + 120583) +119899119872
119872 + 1199012120575+120583
(119911 isin U)
(38)
where 1199012120575+120583 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120583
((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
minus1199012120575+120583100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872 (119911 isin U)
(39)
where the powers are the principal ones
Corollary 13 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) gt 120601 (120583 120575 120574) (119911 isin U) (40)
where 120574 isin [0 1199012120575+120583) and
120601 (120583 120575 120574) = 119896 (120583 120575 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199012120575+120583 minus 120574] 119894119891 120574 isin [0
1199012120575+120583
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199012120575+120583
2 1199012120575+120583)
(41)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
120583
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
gt 120574
(119911 isin U)
(42)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 12 and 13respectively we obtain the following special cases
Corollary 14 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) lt 119901+119899119872
119872 + 1199011205821+1
(119911 isin U)
(43)
where 1199011205821+1 le 119872 then
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
minus1199011205821+1100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(44)
where the powers are the principal ones
Corollary 15 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) gt 120595 (1205821 120574)
(119911 isin U) (45)
where 120574 isin [0 1199011205821+1) and
6 International Journal of Differential Equations
120595 (1205821 120574) = 119896 (1minus1205821 1205821 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199011205821+1 minus 120574] 119894119891 120574 isin [0
1199011205821+1
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199011205821+1
2 1199011205821+1)
(46)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(47)
where the powers are the principal ones
In the next result we will find the relation betweenM119898120572120573120590
119901119899(1205821 120574) andM
119898120572120573120590
119901119899(1 minus 1205821 1205821 120574) For this purpose
taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Theorem 7 we obtain thefollowing result
Corollary 16 Let119891(119911)isinA(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901119899(1205821 984858 (1205821 120572 120573 120574)) (48)
where 120574 isin [0 1199011205821(120572 + 120573119901)) and
984858 (1205821 120572 120573 120574) = 119896 (1minus1205821 1205821 120572 120573 120574) =
119901 minus119899120574
2 [1199011205821 (120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
1199011205821 (120572 + 120573119901)
2]
119901 minus119899 [1199011205821 (120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
1199011205821 (120572 + 120573119901)
2 1199011205821 [(120572 + 120573119901))
(49)
then 119891(119911) isinN119898120572120573120590119901119899
(1 minus 1205821 1205821 120574)
Taking 1205821 = 0 and 119899 = 1 in the above corollary we get thenext special result
Corollary 17 Let119891(119911) isinA(119901)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901(984858 (120572 120573 120574)) (50)
where 120574 isin [0 120572 + 120573119901) and
984858 (120572 120573 120574) = 119896 (1 0 120572 120573 120574)
=
119901 minus120574
2 [(120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
120572 + 120573119901
2]
119901 minus[(120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
120572 + 120573119901
2 (120572 + 120573119901))
(51)
then 119891(119911) isinN119898120572120573120590119901
(1 0 120574)
Again for the special cases of 120583 and 120575 Theorems 6 and 7reduce at once to some results obtained by [4 5]
Remark 18 Taking 119901 = 1 and 119898 = 0 in (7) and 120572 = 1 and120573 = 0 in (10) we get a known result obtained by Irmak et al[5]
Remark 19 Taking119898 = 0 in (7) and 120572 = 1 minus 120573 in (10) we geta known result obtained by Goswami et al [4]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors agreed with the contents of the paper
Acknowledgments
The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia (UKM) under Grant ERGS12013STG06UKM012Thefirst author is also supported byZAMALAHschemegrant under postgraduate center UKM
References
[1] S P Goyal O Singh and P Goswami ldquoSome relations betweencertain classes of analytic multivalent functions involving gen-eralized salagean operatorrdquo Sohag Journal of Mathematics vol1 no 1 pp 27ndash32 2014
[2] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysismdashFifth Romanian-Finnish Seminar Part 1 Proceedingsof the Seminar Held in Bucharest 1981 vol 1013 of Lecture Notesin Mathematics pp 362ndash372 Springer Berlin Germany 1983
[3] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000
International Journal of Differential Equations 7
[4] P Goswami T Bulboaca and S Bansal ldquoSome relationsbetween certain classes of analytic functionsrdquo Journal of Clas-sical Analysis vol 1 no 2 pp 157ndash173 2012
[5] H Irmak T Bulboaca andN Tuneski ldquoSome relations betweencertain classes consisting of 120572-convex type and Bazilevic typefunctionsrdquoAppliedMathematics Letters vol 24 no 12 pp 2010ndash2014 2011
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International Journal of Differential Equations 3
where 119901120575(120572 + 120573119901)120575+120583 le 119872 then
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863120572120573119898120590
119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus119901120575
(120572 + 120573119901)120575+120583
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872 (119911 isin U)
(15)
where the powers are the principal ones
Proof Let the function ℎ(119911) be defined by
ℎ (119911) = (119863120572120573119898120590
119891 (119911)
119911119901)
120583
((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus119901120575
(120572 +120573119901)120575+120583
(16)
From the assumptions 119891 isin A(119901119899)
with(119863120572120573119898120590
119891(119911))(119863120572120573119898120590
119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U wehave that ℎ isinH[0 119899] By a simple manipulation we have
J119898120572120573120590
119901119899(120583 120575) 119891 (119911) = 119901 (120575 + 120583)
+119911ℎ1015840 (119911)
ℎ (119911) + 119901120575 (120572 + 120573119901)120575+120583
(17)
Now letting
120595 (119903 119904 119911) = 119901 (120575 + 120583) +119904
119903 + 119901120575 (120572 + 120573119901)120575+120583
Ω = 119908isinC R (119908) lt 119901 (120575 + 120583)
+119899119872
119872 + 119901120575 (120572 + 120573119901)120575+120583
(18)
we have from (17) and (14) that
120595 (ℎ (119911) 119911ℎ1015840
(119911) 119911) = J119898120572120573120590
119901119899(120583 120575) 119891 (119911) isin Ω
forall119911 isin U
(19)
Further for any 120579 isin R 119870 ge 119899119872 and 119911 isin U since 119872 ge
119901120575(120572 + 120573119901)120575+120583 we also have
R 120595 (119872119890119894120579
119870119890119894120579
119911)
= 119901 (120575 + 120583) +119870R(1
119872+ 119890minus119894120579119901120575 (120572 + 120573119901)120575+120583
)
ge 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (120572 + 120573119901)120575+120583
(119911 isin U)
(20)
which shows that 120595(119872119890119894120579 119870119890119894120579 119911) notin Ω for all 120579 isin R 119870 ge119899119872 and 119911 isin U Therefore according to Lemma 4 we obtain|ℎ(119911)| lt 119872 (119911 isin U) Hence (15) is proven
Theorem7 Let119891(119911) isin A(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 120583 120575 isin R If
R J119898120572120573120590
119901119899(120583 120575) 119891 (119911) gt 119896 (120583 120575 120572 120573 120574)
(119911 isin U)
(21)
where 120574 isin [0 119901120575(120572 + 120573119901)120575+120583) and
119896 (120583 120575 120572 120573 120574) =
119901 (120575 + 120583) minus119899120574
2 [119901120575 (120572 + 120573119901)120575+120583 minus 120574] if 120574 isin [0
119901120575 (120572 + 120573119901)120575+120583
2]
119901 (120575 + 120583) minus119899 [119901120575 (120572 + 120573119901)
120575+120583
minus 120574]
2120574 if 120574 isin [
119901120575 (120572 + 120573119901)
120575+120583
2 119901120575 (120572 + 120573119901)
120575+120583
)
(22)
then 119891 isinN119898120590120572120573119901119899
(120583 120575 120574)
Proof Suppose that
ℎ (119911) =1
119901120575 (120572 + 120573119901)120575+120583
minus 120574
[[
[
(119863120572120573119898
120590119891 (119911)
119911119901)
120583
sdot ((119863120572120573119898120590
119891 (119911))1015840
119911119901minus1)
120575
minus 120574]]
]
(23)
Then ℎ(119911) = 1 + ℎ119899119911119899 + sdot sdot sdot is analytic in U It is easily seen
from (23) that
4 International Journal of Differential Equations
J119898120572120573120590
119901119899(120583 120575) 119891 (119911)
= 119901 (120575 + 120583) +(119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119911ℎ1015840 (119911)
(119901120575 (120572 + 120573119901)120575+120583
minus 120574) ℎ (119911) + 120574
(24)
Further since
120595 (119903 119904 119911) = 119901 (120575 + 120583) +(119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119904
(119901120575 (120572 + 120573119901)120575+120583
minus 120574) 119903 + 120574
Ω = 119908isinC R (119908) gt 119896 (120583 120575 120572 120573 120574)
(25)
it leads to
120595 (ℎ (119911) 119911ℎ1015840
(119911) 119911) = J119898120572120573120590
119901119899(120583 120575) 119891 (119911) isin Ω
forall119911 isin U
(26)
Also for any 119909 isin R 119910 le minus119899(1 + 1199092)2 and 119911 isin U we have
R 120595 (119894119909 119910 119911) = 119901 (120575 + 120583) +120574 (119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119910
[119901120575 (120572 + 120573119901)120575+120583
minus 120574]21199092 + 1205742
le 119901 (120575 + 120583)
minus119899120574 [119901120575 (120572 + 120573119901)
120575+120583
minus 120574]
21 + 1199092
[119901120575 (120572 + 120573119901)120575+120583
minus 120574]21199092 + 1205742
equiv 119902 (119911) le 119896 (120583 120575 120572 120573 120574)
=
lim119909rarrinfin
119902 (119911) if 120574 isin [0119901120575 (120572 + 120573119901)
120575+120583
2]
119902 (0) if 120574 isin [119901120575 (120572 + 120573119901)
120575+120583
2 119901120575 (120572 + 120573119901)
120575+120583
)
(27)
that is 120595(119894119909 119910 119911) notin Ω Finally by Lemma 5 we obtain that119877119890(ℎ(119911)) gt 0 The proof of Theorem 7 is complete
3 Corollaries and Consequences
We will discuss some interesting consequences of the maintheorems that extend some previous results obtained in ([45])
Putting 120572 = 1 120573 = 0 in Theorems 6 and 7 we get thefollowing corollaries
Corollary 8 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also
let 120583 120575 isin R If
R120583119911 (119863119898120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 120575(1+119911 (119863119898
120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840)
lt 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (119911 isin U)
(28)
where 119901120575 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863119898120590119891 (119911)
119911119901)
120583
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120575
minus119901120575
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(29)
where the powers are the principal ones
Corollary 9 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also
let 120583 120575 isin R If
R120583119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 120575(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840)
gt 120593 (120583 120575 120574) (119911 isin U)
(30)
where 120574 isin [0 119901120575) and
120593 (120583 120575 120574) = 119896 (120583 120575 1 0 120574)
=
119901 (120575 + 120583) minus119899120574
2 [119901120575 minus 120574] 119894119891 120574 isin [0
119901120575
2]
119901 (120575 + 120583) minus119899 [119901120575 minus 120574]
2120574 119894119891 120574 isin [
119901120575
2 119901120575)
(31)
then
R
(119863119898
120590119891 (119911)
119911119901)
120583
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120575
gt 120574 (119911 isin U) (32)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 8 and 9respectively we obtain the following special cases
Corollary 10 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898
120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) lt 119901+
119899119872
119872 + 1199011205821
(119911 isin U)
(33)
where 1199011205821 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863119898120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
minus1199011205821
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(34)
where the powers are the principal ones
International Journal of Differential Equations 5
Corollary 11 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) gt 120594 (1205821 120574)
(119911 isin U)
(35)
where 120574 isin [0 1199011205821) and
120594 (1205821 120574) = 119896 (1minus1205821 1205821 1 0 120574)
=
119901 minus119899120574
2 [1199011205821 minus 120574] 119894119891 120574 isin [0
1199011205821
2]
119901 minus119899 [1199011205821 minus 120574]
2120574 119894119891 120574 isin [
11990112058212 1199011205821)
(36)
then
R
(119863119898
120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(37)
where the powers are the principal ones
Next upon taking 120572 = 0 120573 = 1 in Theorems 6 and 7 weobtain the following results
Corollary 12 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) lt 119901 (120575 + 120583) +119899119872
119872 + 1199012120575+120583
(119911 isin U)
(38)
where 1199012120575+120583 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120583
((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
minus1199012120575+120583100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872 (119911 isin U)
(39)
where the powers are the principal ones
Corollary 13 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) gt 120601 (120583 120575 120574) (119911 isin U) (40)
where 120574 isin [0 1199012120575+120583) and
120601 (120583 120575 120574) = 119896 (120583 120575 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199012120575+120583 minus 120574] 119894119891 120574 isin [0
1199012120575+120583
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199012120575+120583
2 1199012120575+120583)
(41)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
120583
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
gt 120574
(119911 isin U)
(42)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 12 and 13respectively we obtain the following special cases
Corollary 14 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) lt 119901+119899119872
119872 + 1199011205821+1
(119911 isin U)
(43)
where 1199011205821+1 le 119872 then
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
minus1199011205821+1100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(44)
where the powers are the principal ones
Corollary 15 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) gt 120595 (1205821 120574)
(119911 isin U) (45)
where 120574 isin [0 1199011205821+1) and
6 International Journal of Differential Equations
120595 (1205821 120574) = 119896 (1minus1205821 1205821 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199011205821+1 minus 120574] 119894119891 120574 isin [0
1199011205821+1
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199011205821+1
2 1199011205821+1)
(46)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(47)
where the powers are the principal ones
In the next result we will find the relation betweenM119898120572120573120590
119901119899(1205821 120574) andM
119898120572120573120590
119901119899(1 minus 1205821 1205821 120574) For this purpose
taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Theorem 7 we obtain thefollowing result
Corollary 16 Let119891(119911)isinA(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901119899(1205821 984858 (1205821 120572 120573 120574)) (48)
where 120574 isin [0 1199011205821(120572 + 120573119901)) and
984858 (1205821 120572 120573 120574) = 119896 (1minus1205821 1205821 120572 120573 120574) =
119901 minus119899120574
2 [1199011205821 (120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
1199011205821 (120572 + 120573119901)
2]
119901 minus119899 [1199011205821 (120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
1199011205821 (120572 + 120573119901)
2 1199011205821 [(120572 + 120573119901))
(49)
then 119891(119911) isinN119898120572120573120590119901119899
(1 minus 1205821 1205821 120574)
Taking 1205821 = 0 and 119899 = 1 in the above corollary we get thenext special result
Corollary 17 Let119891(119911) isinA(119901)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901(984858 (120572 120573 120574)) (50)
where 120574 isin [0 120572 + 120573119901) and
984858 (120572 120573 120574) = 119896 (1 0 120572 120573 120574)
=
119901 minus120574
2 [(120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
120572 + 120573119901
2]
119901 minus[(120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
120572 + 120573119901
2 (120572 + 120573119901))
(51)
then 119891(119911) isinN119898120572120573120590119901
(1 0 120574)
Again for the special cases of 120583 and 120575 Theorems 6 and 7reduce at once to some results obtained by [4 5]
Remark 18 Taking 119901 = 1 and 119898 = 0 in (7) and 120572 = 1 and120573 = 0 in (10) we get a known result obtained by Irmak et al[5]
Remark 19 Taking119898 = 0 in (7) and 120572 = 1 minus 120573 in (10) we geta known result obtained by Goswami et al [4]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors agreed with the contents of the paper
Acknowledgments
The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia (UKM) under Grant ERGS12013STG06UKM012Thefirst author is also supported byZAMALAHschemegrant under postgraduate center UKM
References
[1] S P Goyal O Singh and P Goswami ldquoSome relations betweencertain classes of analytic multivalent functions involving gen-eralized salagean operatorrdquo Sohag Journal of Mathematics vol1 no 1 pp 27ndash32 2014
[2] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysismdashFifth Romanian-Finnish Seminar Part 1 Proceedingsof the Seminar Held in Bucharest 1981 vol 1013 of Lecture Notesin Mathematics pp 362ndash372 Springer Berlin Germany 1983
[3] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000
International Journal of Differential Equations 7
[4] P Goswami T Bulboaca and S Bansal ldquoSome relationsbetween certain classes of analytic functionsrdquo Journal of Clas-sical Analysis vol 1 no 2 pp 157ndash173 2012
[5] H Irmak T Bulboaca andN Tuneski ldquoSome relations betweencertain classes consisting of 120572-convex type and Bazilevic typefunctionsrdquoAppliedMathematics Letters vol 24 no 12 pp 2010ndash2014 2011
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Stochastic AnalysisInternational Journal of
4 International Journal of Differential Equations
J119898120572120573120590
119901119899(120583 120575) 119891 (119911)
= 119901 (120575 + 120583) +(119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119911ℎ1015840 (119911)
(119901120575 (120572 + 120573119901)120575+120583
minus 120574) ℎ (119911) + 120574
(24)
Further since
120595 (119903 119904 119911) = 119901 (120575 + 120583) +(119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119904
(119901120575 (120572 + 120573119901)120575+120583
minus 120574) 119903 + 120574
Ω = 119908isinC R (119908) gt 119896 (120583 120575 120572 120573 120574)
(25)
it leads to
120595 (ℎ (119911) 119911ℎ1015840
(119911) 119911) = J119898120572120573120590
119901119899(120583 120575) 119891 (119911) isin Ω
forall119911 isin U
(26)
Also for any 119909 isin R 119910 le minus119899(1 + 1199092)2 and 119911 isin U we have
R 120595 (119894119909 119910 119911) = 119901 (120575 + 120583) +120574 (119901120575 (120572 + 120573119901)
120575+120583
minus 120574) 119910
[119901120575 (120572 + 120573119901)120575+120583
minus 120574]21199092 + 1205742
le 119901 (120575 + 120583)
minus119899120574 [119901120575 (120572 + 120573119901)
120575+120583
minus 120574]
21 + 1199092
[119901120575 (120572 + 120573119901)120575+120583
minus 120574]21199092 + 1205742
equiv 119902 (119911) le 119896 (120583 120575 120572 120573 120574)
=
lim119909rarrinfin
119902 (119911) if 120574 isin [0119901120575 (120572 + 120573119901)
120575+120583
2]
119902 (0) if 120574 isin [119901120575 (120572 + 120573119901)
120575+120583
2 119901120575 (120572 + 120573119901)
120575+120583
)
(27)
that is 120595(119894119909 119910 119911) notin Ω Finally by Lemma 5 we obtain that119877119890(ℎ(119911)) gt 0 The proof of Theorem 7 is complete
3 Corollaries and Consequences
We will discuss some interesting consequences of the maintheorems that extend some previous results obtained in ([45])
Putting 120572 = 1 120573 = 0 in Theorems 6 and 7 we get thefollowing corollaries
Corollary 8 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also
let 120583 120575 isin R If
R120583119911 (119863119898120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 120575(1+119911 (119863119898
120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840)
lt 119901 (120575 + 120583) +119899119872
119872 + 119901120575 (119911 isin U)
(28)
where 119901120575 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863119898120590119891 (119911)
119911119901)
120583
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120575
minus119901120575
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(29)
where the powers are the principal ones
Corollary 9 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also
let 120583 120575 isin R If
R120583119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 120575(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840)
gt 120593 (120583 120575 120574) (119911 isin U)
(30)
where 120574 isin [0 119901120575) and
120593 (120583 120575 120574) = 119896 (120583 120575 1 0 120574)
=
119901 (120575 + 120583) minus119899120574
2 [119901120575 minus 120574] 119894119891 120574 isin [0
119901120575
2]
119901 (120575 + 120583) minus119899 [119901120575 minus 120574]
2120574 119894119891 120574 isin [
119901120575
2 119901120575)
(31)
then
R
(119863119898
120590119891 (119911)
119911119901)
120583
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120575
gt 120574 (119911 isin U) (32)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 8 and 9respectively we obtain the following special cases
Corollary 10 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898
120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) lt 119901+
119899119872
119872 + 1199011205821
(119911 isin U)
(33)
where 1199011205821 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(119863119898120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
minus1199011205821
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(34)
where the powers are the principal ones
International Journal of Differential Equations 5
Corollary 11 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) gt 120594 (1205821 120574)
(119911 isin U)
(35)
where 120574 isin [0 1199011205821) and
120594 (1205821 120574) = 119896 (1minus1205821 1205821 1 0 120574)
=
119901 minus119899120574
2 [1199011205821 minus 120574] 119894119891 120574 isin [0
1199011205821
2]
119901 minus119899 [1199011205821 minus 120574]
2120574 119894119891 120574 isin [
11990112058212 1199011205821)
(36)
then
R
(119863119898
120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(37)
where the powers are the principal ones
Next upon taking 120572 = 0 120573 = 1 in Theorems 6 and 7 weobtain the following results
Corollary 12 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) lt 119901 (120575 + 120583) +119899119872
119872 + 1199012120575+120583
(119911 isin U)
(38)
where 1199012120575+120583 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120583
((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
minus1199012120575+120583100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872 (119911 isin U)
(39)
where the powers are the principal ones
Corollary 13 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) gt 120601 (120583 120575 120574) (119911 isin U) (40)
where 120574 isin [0 1199012120575+120583) and
120601 (120583 120575 120574) = 119896 (120583 120575 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199012120575+120583 minus 120574] 119894119891 120574 isin [0
1199012120575+120583
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199012120575+120583
2 1199012120575+120583)
(41)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
120583
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
gt 120574
(119911 isin U)
(42)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 12 and 13respectively we obtain the following special cases
Corollary 14 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) lt 119901+119899119872
119872 + 1199011205821+1
(119911 isin U)
(43)
where 1199011205821+1 le 119872 then
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
minus1199011205821+1100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(44)
where the powers are the principal ones
Corollary 15 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) gt 120595 (1205821 120574)
(119911 isin U) (45)
where 120574 isin [0 1199011205821+1) and
6 International Journal of Differential Equations
120595 (1205821 120574) = 119896 (1minus1205821 1205821 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199011205821+1 minus 120574] 119894119891 120574 isin [0
1199011205821+1
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199011205821+1
2 1199011205821+1)
(46)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(47)
where the powers are the principal ones
In the next result we will find the relation betweenM119898120572120573120590
119901119899(1205821 120574) andM
119898120572120573120590
119901119899(1 minus 1205821 1205821 120574) For this purpose
taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Theorem 7 we obtain thefollowing result
Corollary 16 Let119891(119911)isinA(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901119899(1205821 984858 (1205821 120572 120573 120574)) (48)
where 120574 isin [0 1199011205821(120572 + 120573119901)) and
984858 (1205821 120572 120573 120574) = 119896 (1minus1205821 1205821 120572 120573 120574) =
119901 minus119899120574
2 [1199011205821 (120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
1199011205821 (120572 + 120573119901)
2]
119901 minus119899 [1199011205821 (120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
1199011205821 (120572 + 120573119901)
2 1199011205821 [(120572 + 120573119901))
(49)
then 119891(119911) isinN119898120572120573120590119901119899
(1 minus 1205821 1205821 120574)
Taking 1205821 = 0 and 119899 = 1 in the above corollary we get thenext special result
Corollary 17 Let119891(119911) isinA(119901)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901(984858 (120572 120573 120574)) (50)
where 120574 isin [0 120572 + 120573119901) and
984858 (120572 120573 120574) = 119896 (1 0 120572 120573 120574)
=
119901 minus120574
2 [(120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
120572 + 120573119901
2]
119901 minus[(120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
120572 + 120573119901
2 (120572 + 120573119901))
(51)
then 119891(119911) isinN119898120572120573120590119901
(1 0 120574)
Again for the special cases of 120583 and 120575 Theorems 6 and 7reduce at once to some results obtained by [4 5]
Remark 18 Taking 119901 = 1 and 119898 = 0 in (7) and 120572 = 1 and120573 = 0 in (10) we get a known result obtained by Irmak et al[5]
Remark 19 Taking119898 = 0 in (7) and 120572 = 1 minus 120573 in (10) we geta known result obtained by Goswami et al [4]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors agreed with the contents of the paper
Acknowledgments
The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia (UKM) under Grant ERGS12013STG06UKM012Thefirst author is also supported byZAMALAHschemegrant under postgraduate center UKM
References
[1] S P Goyal O Singh and P Goswami ldquoSome relations betweencertain classes of analytic multivalent functions involving gen-eralized salagean operatorrdquo Sohag Journal of Mathematics vol1 no 1 pp 27ndash32 2014
[2] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysismdashFifth Romanian-Finnish Seminar Part 1 Proceedingsof the Seminar Held in Bucharest 1981 vol 1013 of Lecture Notesin Mathematics pp 362ndash372 Springer Berlin Germany 1983
[3] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000
International Journal of Differential Equations 7
[4] P Goswami T Bulboaca and S Bansal ldquoSome relationsbetween certain classes of analytic functionsrdquo Journal of Clas-sical Analysis vol 1 no 2 pp 157ndash173 2012
[5] H Irmak T Bulboaca andN Tuneski ldquoSome relations betweencertain classes consisting of 120572-convex type and Bazilevic typefunctionsrdquoAppliedMathematics Letters vol 24 no 12 pp 2010ndash2014 2011
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Stochastic AnalysisInternational Journal of
International Journal of Differential Equations 5
Corollary 11 Let 119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))(119863119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863119898120590is given by (7) and also let
1205821 isin R If
R(1minus1205821)119911 (119863119898
120590119891 (119911))
1015840
119863119898120590119891 (119911)
+ 1205821(1+119911 (119863119898120590119891 (119911))
10158401015840
(119863119898120590119891 (119911))
1015840) gt 120594 (1205821 120574)
(119911 isin U)
(35)
where 120574 isin [0 1199011205821) and
120594 (1205821 120574) = 119896 (1minus1205821 1205821 1 0 120574)
=
119901 minus119899120574
2 [1199011205821 minus 120574] 119894119891 120574 isin [0
1199011205821
2]
119901 minus119899 [1199011205821 minus 120574]
2120574 119894119891 120574 isin [
11990112058212 1199011205821)
(36)
then
R
(119863119898
120590119891 (119911)
119911119901)
1minus1205821((119863119898120590119891 (119911))
1015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(37)
where the powers are the principal ones
Next upon taking 120572 = 0 120573 = 1 in Theorems 6 and 7 weobtain the following results
Corollary 12 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) lt 119901 (120575 + 120583) +119899119872
119872 + 1199012120575+120583
(119911 isin U)
(38)
where 1199012120575+120583 le 119872 then100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
120583
((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
minus1199012120575+120583100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872 (119911 isin U)
(39)
where the powers are the principal ones
Corollary 13 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 120583 120575 isin R If
R J11989801120590119901119899
(120583 120575) 119891 (119911) gt 120601 (120583 120575 120574) (119911 isin U) (40)
where 120574 isin [0 1199012120575+120583) and
120601 (120583 120575 120574) = 119896 (120583 120575 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199012120575+120583 minus 120574] 119894119891 120574 isin [0
1199012120575+120583
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199012120575+120583
2 1199012120575+120583)
(41)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
120583
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
120575
gt 120574
(119911 isin U)
(42)
where the powers are the principal ones
Taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Corollaries 12 and 13respectively we obtain the following special cases
Corollary 14 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) lt 119901+119899119872
119872 + 1199011205821+1
(119911 isin U)
(43)
where 1199011205821+1 le 119872 then
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
((119863119898120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
minus1199011205821+1100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119872
(119911 isin U)
(44)
where the powers are the principal ones
Corollary 15 Let119891(119911) isin A(119901119899)
with (119863119898120590119891(119911))1015840
[(119863119898120590119891(119911))1015840
+
119911(119863119898120590119891(119911))10158401015840
]1199112(119901minus1) = 0 for all 119911 isin U where 119863119898120590is given by
(7) and also let 1205821 isin R If
R J11989801120590119901119899
(1minus1205821 1205821) 119891 (119911) gt 120595 (1205821 120574)
(119911 isin U) (45)
where 120574 isin [0 1199011205821+1) and
6 International Journal of Differential Equations
120595 (1205821 120574) = 119896 (1minus1205821 1205821 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199011205821+1 minus 120574] 119894119891 120574 isin [0
1199011205821+1
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199011205821+1
2 1199011205821+1)
(46)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(47)
where the powers are the principal ones
In the next result we will find the relation betweenM119898120572120573120590
119901119899(1205821 120574) andM
119898120572120573120590
119901119899(1 minus 1205821 1205821 120574) For this purpose
taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Theorem 7 we obtain thefollowing result
Corollary 16 Let119891(119911)isinA(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901119899(1205821 984858 (1205821 120572 120573 120574)) (48)
where 120574 isin [0 1199011205821(120572 + 120573119901)) and
984858 (1205821 120572 120573 120574) = 119896 (1minus1205821 1205821 120572 120573 120574) =
119901 minus119899120574
2 [1199011205821 (120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
1199011205821 (120572 + 120573119901)
2]
119901 minus119899 [1199011205821 (120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
1199011205821 (120572 + 120573119901)
2 1199011205821 [(120572 + 120573119901))
(49)
then 119891(119911) isinN119898120572120573120590119901119899
(1 minus 1205821 1205821 120574)
Taking 1205821 = 0 and 119899 = 1 in the above corollary we get thenext special result
Corollary 17 Let119891(119911) isinA(119901)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901(984858 (120572 120573 120574)) (50)
where 120574 isin [0 120572 + 120573119901) and
984858 (120572 120573 120574) = 119896 (1 0 120572 120573 120574)
=
119901 minus120574
2 [(120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
120572 + 120573119901
2]
119901 minus[(120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
120572 + 120573119901
2 (120572 + 120573119901))
(51)
then 119891(119911) isinN119898120572120573120590119901
(1 0 120574)
Again for the special cases of 120583 and 120575 Theorems 6 and 7reduce at once to some results obtained by [4 5]
Remark 18 Taking 119901 = 1 and 119898 = 0 in (7) and 120572 = 1 and120573 = 0 in (10) we get a known result obtained by Irmak et al[5]
Remark 19 Taking119898 = 0 in (7) and 120572 = 1 minus 120573 in (10) we geta known result obtained by Goswami et al [4]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors agreed with the contents of the paper
Acknowledgments
The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia (UKM) under Grant ERGS12013STG06UKM012Thefirst author is also supported byZAMALAHschemegrant under postgraduate center UKM
References
[1] S P Goyal O Singh and P Goswami ldquoSome relations betweencertain classes of analytic multivalent functions involving gen-eralized salagean operatorrdquo Sohag Journal of Mathematics vol1 no 1 pp 27ndash32 2014
[2] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysismdashFifth Romanian-Finnish Seminar Part 1 Proceedingsof the Seminar Held in Bucharest 1981 vol 1013 of Lecture Notesin Mathematics pp 362ndash372 Springer Berlin Germany 1983
[3] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000
International Journal of Differential Equations 7
[4] P Goswami T Bulboaca and S Bansal ldquoSome relationsbetween certain classes of analytic functionsrdquo Journal of Clas-sical Analysis vol 1 no 2 pp 157ndash173 2012
[5] H Irmak T Bulboaca andN Tuneski ldquoSome relations betweencertain classes consisting of 120572-convex type and Bazilevic typefunctionsrdquoAppliedMathematics Letters vol 24 no 12 pp 2010ndash2014 2011
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
6 International Journal of Differential Equations
120595 (1205821 120574) = 119896 (1minus1205821 1205821 0 1 120574)
=
119901 (120575 + 120583) minus119899120574
2 [1199011205821+1 minus 120574] 119894119891 120574 isin [0
1199011205821+1
2]
119901 (120575 + 120583) minus119899 [1199012120575+120583 minus 120574]
2120574 119894119891 120574 isin [
1199011205821+1
2 1199011205821+1)
(46)
then
R
((119863119898
120590119891 (119911))
1015840
119911119901minus1)
1minus1205821
sdot ((119863119898120590119891 (119911))
1015840
+ 119911 (119863119898120590119891 (119911))
10158401015840
119911119901minus1)
1205821
gt 120574
(119911 isin U)
(47)
where the powers are the principal ones
In the next result we will find the relation betweenM119898120572120573120590
119901119899(1205821 120574) andM
119898120572120573120590
119901119899(1 minus 1205821 1205821 120574) For this purpose
taking 120583 = 1 minus 1205821 and 120575 = 1205821 in Theorem 7 we obtain thefollowing result
Corollary 16 Let119891(119911)isinA(119901119899)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901119899(1205821 984858 (1205821 120572 120573 120574)) (48)
where 120574 isin [0 1199011205821(120572 + 120573119901)) and
984858 (1205821 120572 120573 120574) = 119896 (1minus1205821 1205821 120572 120573 120574) =
119901 minus119899120574
2 [1199011205821 (120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
1199011205821 (120572 + 120573119901)
2]
119901 minus119899 [1199011205821 (120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
1199011205821 (120572 + 120573119901)
2 1199011205821 [(120572 + 120573119901))
(49)
then 119891(119911) isinN119898120572120573120590119901119899
(1 minus 1205821 1205821 120574)
Taking 1205821 = 0 and 119899 = 1 in the above corollary we get thenext special result
Corollary 17 Let119891(119911) isinA(119901)
with (119863120572120573119898120590119891(119911))(119863120572120573119898
120590119891(119911))1015840
1199112119901minus1 = 0 for all 119911 isin U where 119863120572120573119898120590
is given by (10) and alsolet 1205821 isin R If
119891 (119911) isinM119898120572120573120590
119901(984858 (120572 120573 120574)) (50)
where 120574 isin [0 120572 + 120573119901) and
984858 (120572 120573 120574) = 119896 (1 0 120572 120573 120574)
=
119901 minus120574
2 [(120572 + 120573119901) minus 120574] 119894119891 120574 isin [0
120572 + 120573119901
2]
119901 minus[(120572 + 120573119901) minus 120574]
2120574 119894119891 120574 isin [
120572 + 120573119901
2 (120572 + 120573119901))
(51)
then 119891(119911) isinN119898120572120573120590119901
(1 0 120574)
Again for the special cases of 120583 and 120575 Theorems 6 and 7reduce at once to some results obtained by [4 5]
Remark 18 Taking 119901 = 1 and 119898 = 0 in (7) and 120572 = 1 and120573 = 0 in (10) we get a known result obtained by Irmak et al[5]
Remark 19 Taking119898 = 0 in (7) and 120572 = 1 minus 120573 in (10) we geta known result obtained by Goswami et al [4]
Conflict of Interests
The authors declare that they have no competing interests
Authorsrsquo Contribution
Both authors agreed with the contents of the paper
Acknowledgments
The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia (UKM) under Grant ERGS12013STG06UKM012Thefirst author is also supported byZAMALAHschemegrant under postgraduate center UKM
References
[1] S P Goyal O Singh and P Goswami ldquoSome relations betweencertain classes of analytic multivalent functions involving gen-eralized salagean operatorrdquo Sohag Journal of Mathematics vol1 no 1 pp 27ndash32 2014
[2] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysismdashFifth Romanian-Finnish Seminar Part 1 Proceedingsof the Seminar Held in Bucharest 1981 vol 1013 of Lecture Notesin Mathematics pp 362ndash372 Springer Berlin Germany 1983
[3] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000
International Journal of Differential Equations 7
[4] P Goswami T Bulboaca and S Bansal ldquoSome relationsbetween certain classes of analytic functionsrdquo Journal of Clas-sical Analysis vol 1 no 2 pp 157ndash173 2012
[5] H Irmak T Bulboaca andN Tuneski ldquoSome relations betweencertain classes consisting of 120572-convex type and Bazilevic typefunctionsrdquoAppliedMathematics Letters vol 24 no 12 pp 2010ndash2014 2011
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
International Journal of Differential Equations 7
[4] P Goswami T Bulboaca and S Bansal ldquoSome relationsbetween certain classes of analytic functionsrdquo Journal of Clas-sical Analysis vol 1 no 2 pp 157ndash173 2012
[5] H Irmak T Bulboaca andN Tuneski ldquoSome relations betweencertain classes consisting of 120572-convex type and Bazilevic typefunctionsrdquoAppliedMathematics Letters vol 24 no 12 pp 2010ndash2014 2011
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
![Multivalent Targeting Based Delivery of Therapeutic ...Multivalent Targeting Based Delivery of Therapeutic ... ... 10), . p)]] ...](https://static.fdocuments.net/doc/165x107/5fe28d7a524ece466e32b4fb/multivalent-targeting-based-delivery-of-therapeutic-multivalent-targeting-based.jpg)
