HudaAldwebyandMaslinaDarusdownloads.hindawi.com/journals/aaa/2014/958563.pdf · Research Article...
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Research Article Some Subordination Results on -Analogue of Ruscheweyh Differential Operator Huda Aldweby and Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor (Darul Ehsan), Malaysia Correspondence should be addressed to Maslina Darus; [email protected] Received 5 January 2014; Revised 24 March 2014; Accepted 25 March 2014; Published 14 April 2014 Academic Editor: Sergei V. Pereverzyev Copyright © 2014 H. Aldweby 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 derive the q-analogue of the well-known Ruscheweyh differential operator using the concept of q-derivative. Here, we investigate several interesting properties of this q-operator by making use of the method of differential subordination. 1. Introduction Recently, the area of -analysis has attracted the serious attention of researchers. is great interest is due to its application in various branches of mathematics and physics. e application of -calculus was initiated by Jackson [1, 2]. He was the first to develop -integral and -derivative in a systematic way. Later, from the 80s, geometrical interpre- tation of -analysis has been recognized through studies on quantum groups. It also suggests a relation between integrable systems and -analysis. In ([3–5]) the -analogue of Baskakov Durrmeyer operator has been proposed, which is based on -analogue of beta function. Another impor- tant -generalization of complex operators is -Picard and -Gauss-Weierstrass singular integral operators discussed in [6–8]. e authors studied approximation and geometric properties of these -operators in some subclasses of analytic functions in compact disk. Very recently, other -analogues of differential operators have been introduced in [9]; see also ([10, 11]). ese -operators are defined by using convolu- tion of normalized analytic functions and -hypergeometric functions, where several interesting results are obtained. From this point, it is expected that deriving -analogues of operators defined on the space of analytic functions would be important in future. A comprehensive study on applications of -analysis in operator theory may be found in [12]. We provide some notations and concepts of -calculus used in this paper. All the results can be found in [12–14]. For ∈ N,0<<1, we define [] = 1− 1− , [] !={ [] [ − 1] ⋅ ⋅ ⋅ [1] , = 1, 2, . . . ; 1, = 0. (1) As → 1, [] →, and this is the bookmark of a - analogue: the limit as →1 recovers the classical object. For complex parameters , , , ( ∈ C \ {0, −1, −2, . . .}, || < 1), the -analogue of Gauss’s hypergeometric function 2 Φ 1 (, ; , , ) is defined by 2 Φ 1 (, ; , , ) = ∞ ∑ =0 (, ) (, ) (, ) (, ) , ( ∈ U), (2) where (, ) is the -analogue of Pochhammer symbol defined by (, ) ={ 1, = 0; (1 − ) (1 − ) (1 − 2 ) ⋅ ⋅ ⋅ (1 − −1 ), ∈ N. (3) Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 958563, 6 pages http://dx.doi.org/10.1155/2014/958563
Transcript of HudaAldwebyandMaslinaDarusdownloads.hindawi.com/journals/aaa/2014/958563.pdf · Research Article...
Research ArticleSome Subordination Results on 119902-Analogue of RuscheweyhDifferential Operator
Huda Aldweby and Maslina Darus
School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia43600 Bangi Selangor (Darul Ehsan) Malaysia
Correspondence should be addressed to Maslina Darus maslinaukmmy
Received 5 January 2014 Revised 24 March 2014 Accepted 25 March 2014 Published 14 April 2014
Academic Editor Sergei V Pereverzyev
Copyright copy 2014 H Aldweby and M Darus This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Wederive the q-analogue of thewell-knownRuscheweyh differential operator using the concept of q-derivativeHere we investigateseveral interesting properties of this q-operator by making use of the method of differential subordination
1 Introduction
Recently the area of 119902-analysis has attracted the seriousattention of researchers This great interest is due to itsapplication in various branches of mathematics and physicsThe application of 119902-calculus was initiated by Jackson [1 2]He was the first to develop 119902-integral and 119902-derivative ina systematic way Later from the 80s geometrical interpre-tation of 119902-analysis has been recognized through studieson quantum groups It also suggests a relation betweenintegrable systems and 119902-analysis In ([3ndash5]) the 119902-analogueof Baskakov Durrmeyer operator has been proposed whichis based on 119902-analogue of beta function Another impor-tant 119902-generalization of complex operators is 119902-Picard and119902-Gauss-Weierstrass singular integral operators discussed in[6ndash8] The authors studied approximation and geometricproperties of these 119902-operators in some subclasses of analyticfunctions in compact disk Very recently other 119902-analoguesof differential operators have been introduced in [9] see also([10 11]) These 119902-operators are defined by using convolu-tion of normalized analytic functions and 119902-hypergeometricfunctions where several interesting results are obtainedFrom this point it is expected that deriving 119902-analoguesof operators defined on the space of analytic functionswould be important in future A comprehensive study onapplications of 119902-analysis in operator theory may be found in[12]
We provide some notations and concepts of 119902-calculusused in this paper All the results can be found in [12ndash14] For119899 isin N 0 lt 119902 lt 1 we define
[119899]119902 =1 minus 119902119899
1 minus 119902
[119899]119902 = [119899]119902[119899 minus 1]119902 sdot sdot sdot [1]119902 119899 = 1 2
1 119899 = 0
(1)
As 119902 rarr 1 [119899]119902 rarr 119899 and this is the bookmark of a 119902-analogue the limit as 119902 rarr 1 recovers the classical object
For complex parameters 119886 119887 119888 119902 (119888 isin C 0 minus1
minus2 |119902| lt 1) the 119902-analogue of Gaussrsquos hypergeometricfunction 2Φ1(119886 119887 119888 119902 119911) is defined by
2Φ1 (119886 119887 119888 119902 119911) =
infin
sum
119896=0
(119886 119902)119896(119887 119902)119896
(119902 119902)119896(119888 119902)119896
119911119896 (119911 isin U) (2)
where (119899 119902)119896 is the 119902-analogue of Pochhammer symboldefined by
(119899 119902)119896
= 1 119896 = 0
(1 minus 119899) (1 minus 119899119902) (1 minus 1198991199022) sdot sdot sdot (1 minus 119899119902
119896minus1) 119896 isin N
(3)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 958563 6 pageshttpdxdoiorg1011552014958563
2 Abstract and Applied Analysis
The 119902-derivative of a function ℎ(119909) is defined by
119863119902 (ℎ (119909)) =ℎ (119902119909) minus ℎ (119909)
(119902 minus 1) 119909 119902 = 1 119909 = 0 (4)
and119863119902(ℎ(0)) = 1198911015840(0) For a function ℎ(119911) = 119911119896 observe that
119863119902 (ℎ (119911)) = 119863119902 (119911119896) =
1 minus 119902119896
1 minus 119902119911119896minus1
= [119896]119902119911119896minus1 (5)
then lim119902rarr1119863119902(ℎ(119911)) = lim119902rarr1[119896]119902119911119896minus1
= 119896119911119896minus1
= ℎ1015840(119911)
where ℎ1015840(119911) is the ordinary derivativeNext we state the classA of all functions of the following
form
119891 (119911) = 119911 +
infin
sum
119896=2
119886119896119911119896 (6)
which are analytic in the open unit disk U = 119911 isin C |119911| lt
1 If 119891 and 119892 are analytic functions in U we say that 119891 issubordinate to 119892 written 119891 ≺ 119892 if there is a function 119908
analytic in U with 119908(0) = 0 |119908(119911)| lt 1 for all 119911 isin U suchthat119891(119911) = 119892(119908(119911)) for all 119911 isin U If 119892 is univalent then119891 ≺ 119892
if and only if 119891(0) = 119892(0) and 119891(U) sube 119892(U)For each 119860 and 119861 such that minus1 le 119861 lt 119860 le 1 we define
the function
ℎ (119860 119861 119911) =1 + 119860119911
1 + 119861119911 (119911 isin U) (7)
It is well known that ℎ(119860 119861 119911) for minus1 le 119861 le 1 is theconformal map of the unit disk onto the disk symmetricalwith respect to the real axis having the center (1minus119860119861)(1minus1198612)for 119861 = plusmn 1 and radius (119860 minus 119861)(1 minus 119861
2) The boundary circle
cuts the real axis at the points (1minus119860)(1minus119861) and (1+119860)(1+119861)
Definition 1 Let 119891 isin A Denote by R120582119902the 119902-analogue of
Ruscheweyh operator defined by
R120582
119902119891 (119911) = 119911 +
infin
sum
119896=2
[119896 + 120582 minus 1]119902
[120582]119902[119896 minus 1]119902119886119896119911119896 (8)
where [119886]119902 and [119886]119902 are defined in (1)
From the definition we observe that if 119902 rarr 1 we have
lim119902rarr1
R120582
119902119891 (119911) = 119911 + lim
119902rarr1[
infin
sum
119896=2
[119896 + 120582 minus 1]119902
[120582]119902[119896 minus 1]119902119886119896119911119896]
= 119911 +
infin
sum
119896=2
(119896 + 120582 minus 1)
(120582) (119896 minus 1)119886119896119911119896= R120582119891 (119911)
(9)
where R120582 is Ruscheweyh differential operator which wasdefined in [15] and has been studied by many authors forexample [16ndash18]
It can also be shown that this 119902-operator is hypergeomet-ric in nature as
R120582
119902119891 (119911) = 119911 2Φ1 (119902
120582+1 119902 119902 119902 119911) lowast 119891 (119911) (10)
where 2Φ1 is the 119902-analogue of Gauss hypergeometricfunction defined in (2) and the symbol (lowast) stands for theHadamard product (or convolution)
The following identity is easily verified for the operatorR120582119902
119902120582119911 (119863119902 (R
120582
119902119891 (119911))) = [120582 + 1]119902R
120582+1
119902119891 (119911) minus [120582]119902R
120582
119902119891 (119911)
(11)
2 Main Results
Before we obtain our results we state some known lemmasLet 119875(120573) be the class of functions of the form
120601 (119911) = 1 + 1198881119911 + 11988821199112+ sdot sdot sdot (12)
which are analytic in U and satisfy the following inequality
Re (120601 (119911)) gt 120573 (0 le 120573 lt 1 119911 isin U) (13)
Lemma 2 (see [19]) Let 120601119895 isin 119875(120573119895) be given by (12) where(0 le 120573119895 lt 1 119895 = 1 2) then
(1206011 lowast 1206012) isin 119875 (1205733) (1205733 = 1 minus 2 (1 minus 1205731) (1 minus 1205732)) (14)
and the bound 1205733 is the best possible
Lemma 3 (see [20]) Let the function 120601 given by (12) be inthe class 119875(120573) Then
Re120601 (119911) gt 2120573 minus 1 +2 (1 minus 120573)
1 + |119911| (0 le 120573 lt 1) (15)
Lemma 4 (see [21]) The function (1minus119911)120574 equiv 119890120574 log(1minus119911) 120574 = 0 isunivalent inU if and only if 120574 is either in the closed disk |120574minus1| le1 or in the closed disk |120574 + 1| le 1
We now generalize the lemmas introduced in [22] and[23] respectively using 119902-derivative
Lemma 5 Let ℎ(119911) be analytic and convex univalent inU andℎ(0) = 1 and let 119892(119911) = 1 + 1198871119911 + 11988721199112 + sdot sdot sdot be analytic in U If
119892 (119911) +119911119863119902 (119892 (119911))
119888≺ ℎ (119911) (119911 isin U 119888 = 0) (16)
Then for Re(119888) ge 0
119892 (119911) ≺119888
119911119888int
119911
0
119905119888minus1ℎ (119905) 119889119905 (17)
Proof Suppose that ℎ is analytic and convex univalent in U
and 119892 is analytic in U Letting 119902 rarr 1 in (16) we have
119892 (119911) +1199111198921015840(119911)
119888≺ ℎ (119911) (119911 isin U 119888 = 0) (18)
Then from Lemma in [22] we obtain
119892 (119911) ≺119888
119911119888int
119911
0
119905119888minus1ℎ (119905) 119889119905 (19)
Abstract and Applied Analysis 3
Lemma 6 Let 119902(119911) be univalent in U and let 120579(119908) and 120601(119908)be analytic in a domain119863 containing 119902(U)with 120601(119908) = 0when119908 isin 119902(U) Set119876(119911) = 119911119863119902(119902(119911))120601(119902(119911)) ℎ(119911) = 120579(119902(119911)+119876(119911))and suppose that
(1) Q(z) is starlike univalent in U(2) Re(119911119863119902(ℎ(119911))119876(119911)) = Re((119863119902(120579119902(119911)))120601(119902(119911))) +
(119911119863119902(119876(119911))119876(119911))) gt 0 (119911 isin U)
If 119901(119911) is analytic in U with 119901(0) = 119902(0) 119901(U) sub 119863 and
120579 (119901 (119911)) + 119911119863119902 (119901 (119911)) 120601 (119901 (119911))
≺ 120579 (119902 (119911)) + 119911119863119902 (119902 (119911)) 120601 (119902 (119911)) = ℎ (119911)
(20)
then 119901(119911) ≺ 119902(119911) and 119902(119911) is the best dominant
The proof is similar to the proof of Lemma 5
Theorem 7 Let gt 0 120572 gt 0 and minus1 le 119861 lt 119860 le 1 If 119891 isin Asatisfies
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911≺ ℎ (119860 119861 119911) (21)
then
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906) 119889119906)
1119899
(119899 ge 1)
(22)
The result is sharp
Proof Let
119892 (119911) =
R120582119902119891 (119911)
119911 (23)
for 119891 isin A Then the function 119892(119911) = 1 + 1198871119911 + sdot sdot sdot is analyticin U By using logarithmic 119902-differentiation on both sides of(23) and multiplying by 119911 we have
119911119863119902 (119892 (119911))
119892 (119911)=
119911119877120582
119902119891 (119911)
119877120582119902119891 (119911)
minus 1 (24)
by making use of identity (11) we obtain
119911119863119902 (119892 (119911))
119892 (119911)=[120582 + 1]119902
119902120582
119877120582+1
119902119891 (119911)
119877120582119902119891 (119911)
minus[120582]119902
119902120582minus 1 (25)
Taking into account that [120582 + 1]119902 = [120582]119902 + 119902120582 we obtain
119902120582
[120582 + 1]119902
119911119863119902 (119892 (119911)) + 119892 (119911) =R120582+1119891 (119911)
119911 (26)
From (11) (23) and (26) we get
119892 (119911) +119902120582120572
[120582 + 1]119902
119911119863119902 (119892 (119911)) ≺ ℎ (119860 119861 119911) (27)
Now applying Lemma 5 we have
119892 (119911) ≺[120582 + 1]119902
119902120582120572119911minus[120582+1]119902119902
120582120572int
1
0
119905([120582+1]119902119902
120582120572)minus1
(1 + 119860119905
1 + 119861119905) 119889119905
(28)
or by the concept of subordination
R120582119902119891 (119911)
119911=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + 119860119906119908 (119911)
1 + 119861119906119908 (119911)) 119889119906
(29)
In view of minus1 le 119861 lt 119860 le 1 and 120582 gt 0 it follows from (29)that
Re(R120582119902119891 (119911)
119911) gt
[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906)119889119906
(30)
with the aid of the elementary inequality Re(1199081119899) ge
(Re119908)1119899 for Re119908 gt 0 and 119899 ge 1 Hence inequality (22)follows directly from (30) To show the sharpness of (22) wedefine 119891 isin A by
R120582119902119891 (119911)
119911=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + 119860119906119911
1 + 119861119906119911) 119889119906 (31)
For this function we find that
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911=1 + 119860119911
1 minus 119861119911
R120582119902119891 (119911)
119911997888rarr
[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906) 119889119906
as 119911 997888rarr minus1
(32)
This completes the proof
Corollary 8 Let 119860 = 2120573 minus 1 and 119861 = minus1 where 0 le 120573 lt 1
and 120572 120582 gt 1 If 119891 satisfies
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911≺ ℎ (2120573 minus 1 minus1 119911) (33)
then
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ( (2120573 minus 1) 119906[120582+1]119902119902
120582120572
+2 (1 minus 120573) [120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
1119899
(119899 ge 1)
(34)
4 Abstract and Applied Analysis
Proof Following the same steps as in the proof ofTheorem 7and considering 119892(119911) = R120582
119902119891(119911)119911 the differential subordi-
nation (27) becomes
119892 (119911) +119902120582120572
[120582 + 1]119902
119911119863119902 (119892 (119911)) ≺1 + (2120573 minus 1) 119911
1 + 119911 (35)
Therefore
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + (1 minus 2120573) 119906
1 + 119906)119889119906)
1119899
= ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
times ((2120573 minus 1) +2 (1 minus 120573)
1 + 119906)119889119906)
1119899
= ( (2120573 minus 1) 119906[120582+1]119902119902
120582120572
+2 (1 minus 120573) [120582 + 1]119902
119902120582int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
1119899
(36)
Theorem 9 Let 120582 gt 0 and 0 le 120588 lt 1 Let 120574 be a complexnumber with 120574 = 0 and satisfy either |2120574(1 minus 120588)([120582 + 1]119902119902120582) minus1| le 1 or |2120574(1 minus 120588)([120582 + 1]119902119902120582) + 1| le 1 If 119891 isin A satisfiesthe condition
Re(R120582+1119902119891 (119911)
R120582119902119891 (119911)
) gt 120588 (119911 isin U) (37)
then
(
R120582119902119891 (119911)
119911)
120574
≺1
(1 minus 119911)2120574(1minus120588)([120582+1]119902119902
120582) (119911 isin U) (38)
where 119902(119911) is the best dominant
Proof Let
119901 (119911) = (
R120582119902119891 (119911)
119911)
120574
(119911 isin U) (39)
Then by making use of (11) (37) and (39) we obtain
1 +119902120582119911119863119902 (119901 (119911))
120574[120582 + 1]119902119901 (119911)≺1 + (1 minus 2120588) 119911
1 minus 119911 (119911 isin U) (40)
We now assume that
119902 (119911) =1
(1 minus 119911)2120574(1minus120588)[120582+1]119902119902
120582 120579 (119908) = 1
120601 (119908) =119902120582
120574[120582 + 1]119902119908
(41)
then 119902(119911) is univalent by condition of the theorem andLemma 4 Further it is easy to show that 119902(119911) 120579(119908) and 120601(119908)satisfy the conditions of Lemma 6 Note that the function
119876 (119911) = 119911119863119902 (119902 (119911)) 120601 (119902 (119911)) =2 (1 minus 120588) 119911
1 minus 119911(42)
is univalent starlike in U and
ℎ (119911) = 120579 (119902 (119911)) + 119876 (119911) =1 + (1 minus 2120588) 119911
1 minus 119911 (43)
Combining (40) and Lemma 6 we get the assertion ofTheorem 9
Theorem 10 Let 120572 lt 1 120582 gt 0 and minus1 le 119861119894 lt 119860 119894 le 1 If eachof the functions 119891119894 isin A satisfies the following subordinationcondition
(1 minus 120572)
R120582119902119891119894 (119911)
119911+ 120572
R120582+1119902119891119894 (119911)
119911≺ ℎ (119860 119894 119861119894 119911)
(44)
then
(1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911≺ ℎ (1 minus 2120574 minus1 119911) (45)
where
Θ (119911) = R120582
119902(1198911 lowast 1198912) (119911) (46)
120574 = 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
(47)
Proof we define the function ℎ119894 by
ℎ119894 (119911) = (1 minus 120572)
R120582119902119891119894 (119911)
119911+ 120572
R120582+1119902119891119894 (119911)
119911
(119891119894 isin A 119894 = 1 2)
(48)
we have ℎ119894(119911) isin 119875(120573119894) where 120573119894 = (1 minus119860 119894)(1 minus 119861119894) (119894 = 1 2)By making use of (11) and (48) we obtain
R120582
119902119891119894 (119911) =
[120582 + 1]119902
119902120582120572int
1
0
119905([120582+1]119902119902
120582120572)minus1
ℎ119894 (119905) 119889119905 (119894 = 1 2)
(49)
which in the light of (46) can show that
R120582
119902Θ (119911) =
[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)int
1
0
119905([120582+1]119902119902
120582120572)minus1
ℎ0 (119905) 119889119905
(50)
Abstract and Applied Analysis 5
where for convenience
ℎ0 (119911) = (1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911
=[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)
times int
1
0
119905([120582+1]119902119902
120582120572)minus1
(ℎ1 lowast ℎ2) (119905) 119889119905
(51)
Note that by using Lemma 2we have (ℎ1lowastℎ2) isin 119875(1205733) where1205733 = 1 minus 2(1 minus 1205731)(1 minus 1205732)
Now with an application of Lemma 3 we have
Re (ℎ0 (119911))
=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1 Re ((ℎ1 lowast ℎ2) (119906119911)) 119889119906
ge[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906 |119911|) 119889119906
gt[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906)119889119906
= 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906) = 120574
(52)
which shows that the desired assertion of Theorem 10 holds
Conflict of Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Authorsrsquo Contribution
Huda Aldweby and Maslina Darus read and approved thefinal manuscript
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments made to improve this paper
References
[1] F H Jackson ldquoOn 119902-definite integralsrdquoTheQuarterly Journal ofPure and Applied Mathematics vol 41 pp 193ndash203 1910
[2] F H Jackson ldquoOn 119902-functions and a certain difference opera-torrdquo Transactions of the Royal Society of Edinburgh vol 46 pp253ndash281 1908
[3] A Aral and V Gupta ldquoOn 119902-Baskakov type operatorsrdquoDemon-stratio Mathematica vol 42 no 1 pp 109ndash122 2009
[4] A Aral and V Gupta ldquoGeneralized 119902-Baskakov operatorsrdquoMathematica Slovaca vol 61 no 4 pp 619ndash634 2011
[5] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the 119902-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[6] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of some singular integrals in the unit diskrdquo Journal ofInequalities and Applications Article ID 17231 19 pages 2006
[7] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of generalized singular integrals in the unit diskrdquoJournal of the Korean Mathematical Society vol 43 no 2 pp425ndash443 2006
[8] A Aral ldquoOn the generalized Picard and Gauss Weierstrasssingular integralsrdquo Journal of Computational Analysis andAppli-cations vol 8 no 3 pp 249ndash261 2006
[9] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquo Matematichki Vesnik vol65 no 4 pp 454ndash465 2013
[10] H Aldweby and M Darus ldquoA subclass of harmonic univalentfunctions associated with 119902-analogue of Dziok-Srivastava oper-atorrdquo ISRN Mathematical Analysis vol 2013 Article ID 3823126 pages 2013
[11] H Aldweby and M Darus ldquoOn harmonic meromorphicfunctions associated with basic hypergeometric functionsrdquoTheScientific World Journal vol 2013 Article ID 164287 7 pages2013
[12] A Aral V Gupta and R P Agarwal Applications of q-Calculusin Operator Theory Springer New York NY USA 2013
[13] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[14] H Exton q-Hypergeometric Functions and Applications EllisHorwood Series Mathematics and Its Applications Ellis Hor-wood Chichester UK 1983
[15] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[16] M L Mogra ldquoApplications of Ruscheweyh derivatives andHadamard product to analytic functionsrdquo International Journalof Mathematics and Mathematical Sciences vol 22 no 4 pp795ndash805 1999
[17] K Inayat Noor and S Hussain ldquoOn certain analytic functionsassociated with Ruscheweyh derivatives and bounded Mocanuvariationrdquo Journal of Mathematical Analysis and Applicationsvol 340 no 2 pp 1145ndash1152 2008
[18] S L Shukla and V Kumar ldquoUnivalent functions defined byRuscheweyh derivativesrdquo International Journal of Mathematicsand Mathematical Sciences vol 6 no 3 pp 483ndash486 1983
[19] G S Rao and R Saravanan ldquoSome results concerning bestuniform coap- proximationrdquo Journal of Inequalities in Pure andApplied Math vol 3 no 2 p 24 2002
[20] G S Rao and K R Chandrasekaran ldquoCharacterization ofelements of best coapproximation in normed linear spacesrdquoPure and Applied Mathematical Sciences vol 26 pp 139ndash1471987
[21] M S Robertson ldquoCertain classes of starlike functionsrdquo TheMichiganMathematical Journal vol 32 no 2 pp 135ndash140 1985
6 Abstract and Applied Analysis
[22] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981
[23] S S Miller and P T Mocanu ldquoOn some classes of first-order differential subordinationsrdquo The Michigan MathematicalJournal vol 32 no 2 pp 185ndash195 1985
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
The 119902-derivative of a function ℎ(119909) is defined by
119863119902 (ℎ (119909)) =ℎ (119902119909) minus ℎ (119909)
(119902 minus 1) 119909 119902 = 1 119909 = 0 (4)
and119863119902(ℎ(0)) = 1198911015840(0) For a function ℎ(119911) = 119911119896 observe that
119863119902 (ℎ (119911)) = 119863119902 (119911119896) =
1 minus 119902119896
1 minus 119902119911119896minus1
= [119896]119902119911119896minus1 (5)
then lim119902rarr1119863119902(ℎ(119911)) = lim119902rarr1[119896]119902119911119896minus1
= 119896119911119896minus1
= ℎ1015840(119911)
where ℎ1015840(119911) is the ordinary derivativeNext we state the classA of all functions of the following
form
119891 (119911) = 119911 +
infin
sum
119896=2
119886119896119911119896 (6)
which are analytic in the open unit disk U = 119911 isin C |119911| lt
1 If 119891 and 119892 are analytic functions in U we say that 119891 issubordinate to 119892 written 119891 ≺ 119892 if there is a function 119908
analytic in U with 119908(0) = 0 |119908(119911)| lt 1 for all 119911 isin U suchthat119891(119911) = 119892(119908(119911)) for all 119911 isin U If 119892 is univalent then119891 ≺ 119892
if and only if 119891(0) = 119892(0) and 119891(U) sube 119892(U)For each 119860 and 119861 such that minus1 le 119861 lt 119860 le 1 we define
the function
ℎ (119860 119861 119911) =1 + 119860119911
1 + 119861119911 (119911 isin U) (7)
It is well known that ℎ(119860 119861 119911) for minus1 le 119861 le 1 is theconformal map of the unit disk onto the disk symmetricalwith respect to the real axis having the center (1minus119860119861)(1minus1198612)for 119861 = plusmn 1 and radius (119860 minus 119861)(1 minus 119861
2) The boundary circle
cuts the real axis at the points (1minus119860)(1minus119861) and (1+119860)(1+119861)
Definition 1 Let 119891 isin A Denote by R120582119902the 119902-analogue of
Ruscheweyh operator defined by
R120582
119902119891 (119911) = 119911 +
infin
sum
119896=2
[119896 + 120582 minus 1]119902
[120582]119902[119896 minus 1]119902119886119896119911119896 (8)
where [119886]119902 and [119886]119902 are defined in (1)
From the definition we observe that if 119902 rarr 1 we have
lim119902rarr1
R120582
119902119891 (119911) = 119911 + lim
119902rarr1[
infin
sum
119896=2
[119896 + 120582 minus 1]119902
[120582]119902[119896 minus 1]119902119886119896119911119896]
= 119911 +
infin
sum
119896=2
(119896 + 120582 minus 1)
(120582) (119896 minus 1)119886119896119911119896= R120582119891 (119911)
(9)
where R120582 is Ruscheweyh differential operator which wasdefined in [15] and has been studied by many authors forexample [16ndash18]
It can also be shown that this 119902-operator is hypergeomet-ric in nature as
R120582
119902119891 (119911) = 119911 2Φ1 (119902
120582+1 119902 119902 119902 119911) lowast 119891 (119911) (10)
where 2Φ1 is the 119902-analogue of Gauss hypergeometricfunction defined in (2) and the symbol (lowast) stands for theHadamard product (or convolution)
The following identity is easily verified for the operatorR120582119902
119902120582119911 (119863119902 (R
120582
119902119891 (119911))) = [120582 + 1]119902R
120582+1
119902119891 (119911) minus [120582]119902R
120582
119902119891 (119911)
(11)
2 Main Results
Before we obtain our results we state some known lemmasLet 119875(120573) be the class of functions of the form
120601 (119911) = 1 + 1198881119911 + 11988821199112+ sdot sdot sdot (12)
which are analytic in U and satisfy the following inequality
Re (120601 (119911)) gt 120573 (0 le 120573 lt 1 119911 isin U) (13)
Lemma 2 (see [19]) Let 120601119895 isin 119875(120573119895) be given by (12) where(0 le 120573119895 lt 1 119895 = 1 2) then
(1206011 lowast 1206012) isin 119875 (1205733) (1205733 = 1 minus 2 (1 minus 1205731) (1 minus 1205732)) (14)
and the bound 1205733 is the best possible
Lemma 3 (see [20]) Let the function 120601 given by (12) be inthe class 119875(120573) Then
Re120601 (119911) gt 2120573 minus 1 +2 (1 minus 120573)
1 + |119911| (0 le 120573 lt 1) (15)
Lemma 4 (see [21]) The function (1minus119911)120574 equiv 119890120574 log(1minus119911) 120574 = 0 isunivalent inU if and only if 120574 is either in the closed disk |120574minus1| le1 or in the closed disk |120574 + 1| le 1
We now generalize the lemmas introduced in [22] and[23] respectively using 119902-derivative
Lemma 5 Let ℎ(119911) be analytic and convex univalent inU andℎ(0) = 1 and let 119892(119911) = 1 + 1198871119911 + 11988721199112 + sdot sdot sdot be analytic in U If
119892 (119911) +119911119863119902 (119892 (119911))
119888≺ ℎ (119911) (119911 isin U 119888 = 0) (16)
Then for Re(119888) ge 0
119892 (119911) ≺119888
119911119888int
119911
0
119905119888minus1ℎ (119905) 119889119905 (17)
Proof Suppose that ℎ is analytic and convex univalent in U
and 119892 is analytic in U Letting 119902 rarr 1 in (16) we have
119892 (119911) +1199111198921015840(119911)
119888≺ ℎ (119911) (119911 isin U 119888 = 0) (18)
Then from Lemma in [22] we obtain
119892 (119911) ≺119888
119911119888int
119911
0
119905119888minus1ℎ (119905) 119889119905 (19)
Abstract and Applied Analysis 3
Lemma 6 Let 119902(119911) be univalent in U and let 120579(119908) and 120601(119908)be analytic in a domain119863 containing 119902(U)with 120601(119908) = 0when119908 isin 119902(U) Set119876(119911) = 119911119863119902(119902(119911))120601(119902(119911)) ℎ(119911) = 120579(119902(119911)+119876(119911))and suppose that
(1) Q(z) is starlike univalent in U(2) Re(119911119863119902(ℎ(119911))119876(119911)) = Re((119863119902(120579119902(119911)))120601(119902(119911))) +
(119911119863119902(119876(119911))119876(119911))) gt 0 (119911 isin U)
If 119901(119911) is analytic in U with 119901(0) = 119902(0) 119901(U) sub 119863 and
120579 (119901 (119911)) + 119911119863119902 (119901 (119911)) 120601 (119901 (119911))
≺ 120579 (119902 (119911)) + 119911119863119902 (119902 (119911)) 120601 (119902 (119911)) = ℎ (119911)
(20)
then 119901(119911) ≺ 119902(119911) and 119902(119911) is the best dominant
The proof is similar to the proof of Lemma 5
Theorem 7 Let gt 0 120572 gt 0 and minus1 le 119861 lt 119860 le 1 If 119891 isin Asatisfies
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911≺ ℎ (119860 119861 119911) (21)
then
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906) 119889119906)
1119899
(119899 ge 1)
(22)
The result is sharp
Proof Let
119892 (119911) =
R120582119902119891 (119911)
119911 (23)
for 119891 isin A Then the function 119892(119911) = 1 + 1198871119911 + sdot sdot sdot is analyticin U By using logarithmic 119902-differentiation on both sides of(23) and multiplying by 119911 we have
119911119863119902 (119892 (119911))
119892 (119911)=
119911119877120582
119902119891 (119911)
119877120582119902119891 (119911)
minus 1 (24)
by making use of identity (11) we obtain
119911119863119902 (119892 (119911))
119892 (119911)=[120582 + 1]119902
119902120582
119877120582+1
119902119891 (119911)
119877120582119902119891 (119911)
minus[120582]119902
119902120582minus 1 (25)
Taking into account that [120582 + 1]119902 = [120582]119902 + 119902120582 we obtain
119902120582
[120582 + 1]119902
119911119863119902 (119892 (119911)) + 119892 (119911) =R120582+1119891 (119911)
119911 (26)
From (11) (23) and (26) we get
119892 (119911) +119902120582120572
[120582 + 1]119902
119911119863119902 (119892 (119911)) ≺ ℎ (119860 119861 119911) (27)
Now applying Lemma 5 we have
119892 (119911) ≺[120582 + 1]119902
119902120582120572119911minus[120582+1]119902119902
120582120572int
1
0
119905([120582+1]119902119902
120582120572)minus1
(1 + 119860119905
1 + 119861119905) 119889119905
(28)
or by the concept of subordination
R120582119902119891 (119911)
119911=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + 119860119906119908 (119911)
1 + 119861119906119908 (119911)) 119889119906
(29)
In view of minus1 le 119861 lt 119860 le 1 and 120582 gt 0 it follows from (29)that
Re(R120582119902119891 (119911)
119911) gt
[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906)119889119906
(30)
with the aid of the elementary inequality Re(1199081119899) ge
(Re119908)1119899 for Re119908 gt 0 and 119899 ge 1 Hence inequality (22)follows directly from (30) To show the sharpness of (22) wedefine 119891 isin A by
R120582119902119891 (119911)
119911=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + 119860119906119911
1 + 119861119906119911) 119889119906 (31)
For this function we find that
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911=1 + 119860119911
1 minus 119861119911
R120582119902119891 (119911)
119911997888rarr
[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906) 119889119906
as 119911 997888rarr minus1
(32)
This completes the proof
Corollary 8 Let 119860 = 2120573 minus 1 and 119861 = minus1 where 0 le 120573 lt 1
and 120572 120582 gt 1 If 119891 satisfies
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911≺ ℎ (2120573 minus 1 minus1 119911) (33)
then
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ( (2120573 minus 1) 119906[120582+1]119902119902
120582120572
+2 (1 minus 120573) [120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
1119899
(119899 ge 1)
(34)
4 Abstract and Applied Analysis
Proof Following the same steps as in the proof ofTheorem 7and considering 119892(119911) = R120582
119902119891(119911)119911 the differential subordi-
nation (27) becomes
119892 (119911) +119902120582120572
[120582 + 1]119902
119911119863119902 (119892 (119911)) ≺1 + (2120573 minus 1) 119911
1 + 119911 (35)
Therefore
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + (1 minus 2120573) 119906
1 + 119906)119889119906)
1119899
= ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
times ((2120573 minus 1) +2 (1 minus 120573)
1 + 119906)119889119906)
1119899
= ( (2120573 minus 1) 119906[120582+1]119902119902
120582120572
+2 (1 minus 120573) [120582 + 1]119902
119902120582int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
1119899
(36)
Theorem 9 Let 120582 gt 0 and 0 le 120588 lt 1 Let 120574 be a complexnumber with 120574 = 0 and satisfy either |2120574(1 minus 120588)([120582 + 1]119902119902120582) minus1| le 1 or |2120574(1 minus 120588)([120582 + 1]119902119902120582) + 1| le 1 If 119891 isin A satisfiesthe condition
Re(R120582+1119902119891 (119911)
R120582119902119891 (119911)
) gt 120588 (119911 isin U) (37)
then
(
R120582119902119891 (119911)
119911)
120574
≺1
(1 minus 119911)2120574(1minus120588)([120582+1]119902119902
120582) (119911 isin U) (38)
where 119902(119911) is the best dominant
Proof Let
119901 (119911) = (
R120582119902119891 (119911)
119911)
120574
(119911 isin U) (39)
Then by making use of (11) (37) and (39) we obtain
1 +119902120582119911119863119902 (119901 (119911))
120574[120582 + 1]119902119901 (119911)≺1 + (1 minus 2120588) 119911
1 minus 119911 (119911 isin U) (40)
We now assume that
119902 (119911) =1
(1 minus 119911)2120574(1minus120588)[120582+1]119902119902
120582 120579 (119908) = 1
120601 (119908) =119902120582
120574[120582 + 1]119902119908
(41)
then 119902(119911) is univalent by condition of the theorem andLemma 4 Further it is easy to show that 119902(119911) 120579(119908) and 120601(119908)satisfy the conditions of Lemma 6 Note that the function
119876 (119911) = 119911119863119902 (119902 (119911)) 120601 (119902 (119911)) =2 (1 minus 120588) 119911
1 minus 119911(42)
is univalent starlike in U and
ℎ (119911) = 120579 (119902 (119911)) + 119876 (119911) =1 + (1 minus 2120588) 119911
1 minus 119911 (43)
Combining (40) and Lemma 6 we get the assertion ofTheorem 9
Theorem 10 Let 120572 lt 1 120582 gt 0 and minus1 le 119861119894 lt 119860 119894 le 1 If eachof the functions 119891119894 isin A satisfies the following subordinationcondition
(1 minus 120572)
R120582119902119891119894 (119911)
119911+ 120572
R120582+1119902119891119894 (119911)
119911≺ ℎ (119860 119894 119861119894 119911)
(44)
then
(1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911≺ ℎ (1 minus 2120574 minus1 119911) (45)
where
Θ (119911) = R120582
119902(1198911 lowast 1198912) (119911) (46)
120574 = 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
(47)
Proof we define the function ℎ119894 by
ℎ119894 (119911) = (1 minus 120572)
R120582119902119891119894 (119911)
119911+ 120572
R120582+1119902119891119894 (119911)
119911
(119891119894 isin A 119894 = 1 2)
(48)
we have ℎ119894(119911) isin 119875(120573119894) where 120573119894 = (1 minus119860 119894)(1 minus 119861119894) (119894 = 1 2)By making use of (11) and (48) we obtain
R120582
119902119891119894 (119911) =
[120582 + 1]119902
119902120582120572int
1
0
119905([120582+1]119902119902
120582120572)minus1
ℎ119894 (119905) 119889119905 (119894 = 1 2)
(49)
which in the light of (46) can show that
R120582
119902Θ (119911) =
[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)int
1
0
119905([120582+1]119902119902
120582120572)minus1
ℎ0 (119905) 119889119905
(50)
Abstract and Applied Analysis 5
where for convenience
ℎ0 (119911) = (1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911
=[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)
times int
1
0
119905([120582+1]119902119902
120582120572)minus1
(ℎ1 lowast ℎ2) (119905) 119889119905
(51)
Note that by using Lemma 2we have (ℎ1lowastℎ2) isin 119875(1205733) where1205733 = 1 minus 2(1 minus 1205731)(1 minus 1205732)
Now with an application of Lemma 3 we have
Re (ℎ0 (119911))
=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1 Re ((ℎ1 lowast ℎ2) (119906119911)) 119889119906
ge[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906 |119911|) 119889119906
gt[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906)119889119906
= 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906) = 120574
(52)
which shows that the desired assertion of Theorem 10 holds
Conflict of Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Authorsrsquo Contribution
Huda Aldweby and Maslina Darus read and approved thefinal manuscript
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments made to improve this paper
References
[1] F H Jackson ldquoOn 119902-definite integralsrdquoTheQuarterly Journal ofPure and Applied Mathematics vol 41 pp 193ndash203 1910
[2] F H Jackson ldquoOn 119902-functions and a certain difference opera-torrdquo Transactions of the Royal Society of Edinburgh vol 46 pp253ndash281 1908
[3] A Aral and V Gupta ldquoOn 119902-Baskakov type operatorsrdquoDemon-stratio Mathematica vol 42 no 1 pp 109ndash122 2009
[4] A Aral and V Gupta ldquoGeneralized 119902-Baskakov operatorsrdquoMathematica Slovaca vol 61 no 4 pp 619ndash634 2011
[5] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the 119902-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[6] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of some singular integrals in the unit diskrdquo Journal ofInequalities and Applications Article ID 17231 19 pages 2006
[7] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of generalized singular integrals in the unit diskrdquoJournal of the Korean Mathematical Society vol 43 no 2 pp425ndash443 2006
[8] A Aral ldquoOn the generalized Picard and Gauss Weierstrasssingular integralsrdquo Journal of Computational Analysis andAppli-cations vol 8 no 3 pp 249ndash261 2006
[9] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquo Matematichki Vesnik vol65 no 4 pp 454ndash465 2013
[10] H Aldweby and M Darus ldquoA subclass of harmonic univalentfunctions associated with 119902-analogue of Dziok-Srivastava oper-atorrdquo ISRN Mathematical Analysis vol 2013 Article ID 3823126 pages 2013
[11] H Aldweby and M Darus ldquoOn harmonic meromorphicfunctions associated with basic hypergeometric functionsrdquoTheScientific World Journal vol 2013 Article ID 164287 7 pages2013
[12] A Aral V Gupta and R P Agarwal Applications of q-Calculusin Operator Theory Springer New York NY USA 2013
[13] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[14] H Exton q-Hypergeometric Functions and Applications EllisHorwood Series Mathematics and Its Applications Ellis Hor-wood Chichester UK 1983
[15] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[16] M L Mogra ldquoApplications of Ruscheweyh derivatives andHadamard product to analytic functionsrdquo International Journalof Mathematics and Mathematical Sciences vol 22 no 4 pp795ndash805 1999
[17] K Inayat Noor and S Hussain ldquoOn certain analytic functionsassociated with Ruscheweyh derivatives and bounded Mocanuvariationrdquo Journal of Mathematical Analysis and Applicationsvol 340 no 2 pp 1145ndash1152 2008
[18] S L Shukla and V Kumar ldquoUnivalent functions defined byRuscheweyh derivativesrdquo International Journal of Mathematicsand Mathematical Sciences vol 6 no 3 pp 483ndash486 1983
[19] G S Rao and R Saravanan ldquoSome results concerning bestuniform coap- proximationrdquo Journal of Inequalities in Pure andApplied Math vol 3 no 2 p 24 2002
[20] G S Rao and K R Chandrasekaran ldquoCharacterization ofelements of best coapproximation in normed linear spacesrdquoPure and Applied Mathematical Sciences vol 26 pp 139ndash1471987
[21] M S Robertson ldquoCertain classes of starlike functionsrdquo TheMichiganMathematical Journal vol 32 no 2 pp 135ndash140 1985
6 Abstract and Applied Analysis
[22] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981
[23] S S Miller and P T Mocanu ldquoOn some classes of first-order differential subordinationsrdquo The Michigan MathematicalJournal vol 32 no 2 pp 185ndash195 1985
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
Abstract and Applied Analysis 3
Lemma 6 Let 119902(119911) be univalent in U and let 120579(119908) and 120601(119908)be analytic in a domain119863 containing 119902(U)with 120601(119908) = 0when119908 isin 119902(U) Set119876(119911) = 119911119863119902(119902(119911))120601(119902(119911)) ℎ(119911) = 120579(119902(119911)+119876(119911))and suppose that
(1) Q(z) is starlike univalent in U(2) Re(119911119863119902(ℎ(119911))119876(119911)) = Re((119863119902(120579119902(119911)))120601(119902(119911))) +
(119911119863119902(119876(119911))119876(119911))) gt 0 (119911 isin U)
If 119901(119911) is analytic in U with 119901(0) = 119902(0) 119901(U) sub 119863 and
120579 (119901 (119911)) + 119911119863119902 (119901 (119911)) 120601 (119901 (119911))
≺ 120579 (119902 (119911)) + 119911119863119902 (119902 (119911)) 120601 (119902 (119911)) = ℎ (119911)
(20)
then 119901(119911) ≺ 119902(119911) and 119902(119911) is the best dominant
The proof is similar to the proof of Lemma 5
Theorem 7 Let gt 0 120572 gt 0 and minus1 le 119861 lt 119860 le 1 If 119891 isin Asatisfies
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911≺ ℎ (119860 119861 119911) (21)
then
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906) 119889119906)
1119899
(119899 ge 1)
(22)
The result is sharp
Proof Let
119892 (119911) =
R120582119902119891 (119911)
119911 (23)
for 119891 isin A Then the function 119892(119911) = 1 + 1198871119911 + sdot sdot sdot is analyticin U By using logarithmic 119902-differentiation on both sides of(23) and multiplying by 119911 we have
119911119863119902 (119892 (119911))
119892 (119911)=
119911119877120582
119902119891 (119911)
119877120582119902119891 (119911)
minus 1 (24)
by making use of identity (11) we obtain
119911119863119902 (119892 (119911))
119892 (119911)=[120582 + 1]119902
119902120582
119877120582+1
119902119891 (119911)
119877120582119902119891 (119911)
minus[120582]119902
119902120582minus 1 (25)
Taking into account that [120582 + 1]119902 = [120582]119902 + 119902120582 we obtain
119902120582
[120582 + 1]119902
119911119863119902 (119892 (119911)) + 119892 (119911) =R120582+1119891 (119911)
119911 (26)
From (11) (23) and (26) we get
119892 (119911) +119902120582120572
[120582 + 1]119902
119911119863119902 (119892 (119911)) ≺ ℎ (119860 119861 119911) (27)
Now applying Lemma 5 we have
119892 (119911) ≺[120582 + 1]119902
119902120582120572119911minus[120582+1]119902119902
120582120572int
1
0
119905([120582+1]119902119902
120582120572)minus1
(1 + 119860119905
1 + 119861119905) 119889119905
(28)
or by the concept of subordination
R120582119902119891 (119911)
119911=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + 119860119906119908 (119911)
1 + 119861119906119908 (119911)) 119889119906
(29)
In view of minus1 le 119861 lt 119860 le 1 and 120582 gt 0 it follows from (29)that
Re(R120582119902119891 (119911)
119911) gt
[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906)119889119906
(30)
with the aid of the elementary inequality Re(1199081119899) ge
(Re119908)1119899 for Re119908 gt 0 and 119899 ge 1 Hence inequality (22)follows directly from (30) To show the sharpness of (22) wedefine 119891 isin A by
R120582119902119891 (119911)
119911=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + 119860119906119911
1 + 119861119906119911) 119889119906 (31)
For this function we find that
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911=1 + 119860119911
1 minus 119861119911
R120582119902119891 (119911)
119911997888rarr
[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 minus 119860119906
1 minus 119861119906) 119889119906
as 119911 997888rarr minus1
(32)
This completes the proof
Corollary 8 Let 119860 = 2120573 minus 1 and 119861 = minus1 where 0 le 120573 lt 1
and 120572 120582 gt 1 If 119891 satisfies
(1 minus 120572)
R120582119902119891 (119911)
119911+ 120572
R120582+1119902119891 (119911)
119911≺ ℎ (2120573 minus 1 minus1 119911) (33)
then
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ( (2120573 minus 1) 119906[120582+1]119902119902
120582120572
+2 (1 minus 120573) [120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
1119899
(119899 ge 1)
(34)
4 Abstract and Applied Analysis
Proof Following the same steps as in the proof ofTheorem 7and considering 119892(119911) = R120582
119902119891(119911)119911 the differential subordi-
nation (27) becomes
119892 (119911) +119902120582120572
[120582 + 1]119902
119911119863119902 (119892 (119911)) ≺1 + (2120573 minus 1) 119911
1 + 119911 (35)
Therefore
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + (1 minus 2120573) 119906
1 + 119906)119889119906)
1119899
= ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
times ((2120573 minus 1) +2 (1 minus 120573)
1 + 119906)119889119906)
1119899
= ( (2120573 minus 1) 119906[120582+1]119902119902
120582120572
+2 (1 minus 120573) [120582 + 1]119902
119902120582int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
1119899
(36)
Theorem 9 Let 120582 gt 0 and 0 le 120588 lt 1 Let 120574 be a complexnumber with 120574 = 0 and satisfy either |2120574(1 minus 120588)([120582 + 1]119902119902120582) minus1| le 1 or |2120574(1 minus 120588)([120582 + 1]119902119902120582) + 1| le 1 If 119891 isin A satisfiesthe condition
Re(R120582+1119902119891 (119911)
R120582119902119891 (119911)
) gt 120588 (119911 isin U) (37)
then
(
R120582119902119891 (119911)
119911)
120574
≺1
(1 minus 119911)2120574(1minus120588)([120582+1]119902119902
120582) (119911 isin U) (38)
where 119902(119911) is the best dominant
Proof Let
119901 (119911) = (
R120582119902119891 (119911)
119911)
120574
(119911 isin U) (39)
Then by making use of (11) (37) and (39) we obtain
1 +119902120582119911119863119902 (119901 (119911))
120574[120582 + 1]119902119901 (119911)≺1 + (1 minus 2120588) 119911
1 minus 119911 (119911 isin U) (40)
We now assume that
119902 (119911) =1
(1 minus 119911)2120574(1minus120588)[120582+1]119902119902
120582 120579 (119908) = 1
120601 (119908) =119902120582
120574[120582 + 1]119902119908
(41)
then 119902(119911) is univalent by condition of the theorem andLemma 4 Further it is easy to show that 119902(119911) 120579(119908) and 120601(119908)satisfy the conditions of Lemma 6 Note that the function
119876 (119911) = 119911119863119902 (119902 (119911)) 120601 (119902 (119911)) =2 (1 minus 120588) 119911
1 minus 119911(42)
is univalent starlike in U and
ℎ (119911) = 120579 (119902 (119911)) + 119876 (119911) =1 + (1 minus 2120588) 119911
1 minus 119911 (43)
Combining (40) and Lemma 6 we get the assertion ofTheorem 9
Theorem 10 Let 120572 lt 1 120582 gt 0 and minus1 le 119861119894 lt 119860 119894 le 1 If eachof the functions 119891119894 isin A satisfies the following subordinationcondition
(1 minus 120572)
R120582119902119891119894 (119911)
119911+ 120572
R120582+1119902119891119894 (119911)
119911≺ ℎ (119860 119894 119861119894 119911)
(44)
then
(1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911≺ ℎ (1 minus 2120574 minus1 119911) (45)
where
Θ (119911) = R120582
119902(1198911 lowast 1198912) (119911) (46)
120574 = 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
(47)
Proof we define the function ℎ119894 by
ℎ119894 (119911) = (1 minus 120572)
R120582119902119891119894 (119911)
119911+ 120572
R120582+1119902119891119894 (119911)
119911
(119891119894 isin A 119894 = 1 2)
(48)
we have ℎ119894(119911) isin 119875(120573119894) where 120573119894 = (1 minus119860 119894)(1 minus 119861119894) (119894 = 1 2)By making use of (11) and (48) we obtain
R120582
119902119891119894 (119911) =
[120582 + 1]119902
119902120582120572int
1
0
119905([120582+1]119902119902
120582120572)minus1
ℎ119894 (119905) 119889119905 (119894 = 1 2)
(49)
which in the light of (46) can show that
R120582
119902Θ (119911) =
[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)int
1
0
119905([120582+1]119902119902
120582120572)minus1
ℎ0 (119905) 119889119905
(50)
Abstract and Applied Analysis 5
where for convenience
ℎ0 (119911) = (1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911
=[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)
times int
1
0
119905([120582+1]119902119902
120582120572)minus1
(ℎ1 lowast ℎ2) (119905) 119889119905
(51)
Note that by using Lemma 2we have (ℎ1lowastℎ2) isin 119875(1205733) where1205733 = 1 minus 2(1 minus 1205731)(1 minus 1205732)
Now with an application of Lemma 3 we have
Re (ℎ0 (119911))
=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1 Re ((ℎ1 lowast ℎ2) (119906119911)) 119889119906
ge[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906 |119911|) 119889119906
gt[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906)119889119906
= 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906) = 120574
(52)
which shows that the desired assertion of Theorem 10 holds
Conflict of Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Authorsrsquo Contribution
Huda Aldweby and Maslina Darus read and approved thefinal manuscript
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments made to improve this paper
References
[1] F H Jackson ldquoOn 119902-definite integralsrdquoTheQuarterly Journal ofPure and Applied Mathematics vol 41 pp 193ndash203 1910
[2] F H Jackson ldquoOn 119902-functions and a certain difference opera-torrdquo Transactions of the Royal Society of Edinburgh vol 46 pp253ndash281 1908
[3] A Aral and V Gupta ldquoOn 119902-Baskakov type operatorsrdquoDemon-stratio Mathematica vol 42 no 1 pp 109ndash122 2009
[4] A Aral and V Gupta ldquoGeneralized 119902-Baskakov operatorsrdquoMathematica Slovaca vol 61 no 4 pp 619ndash634 2011
[5] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the 119902-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[6] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of some singular integrals in the unit diskrdquo Journal ofInequalities and Applications Article ID 17231 19 pages 2006
[7] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of generalized singular integrals in the unit diskrdquoJournal of the Korean Mathematical Society vol 43 no 2 pp425ndash443 2006
[8] A Aral ldquoOn the generalized Picard and Gauss Weierstrasssingular integralsrdquo Journal of Computational Analysis andAppli-cations vol 8 no 3 pp 249ndash261 2006
[9] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquo Matematichki Vesnik vol65 no 4 pp 454ndash465 2013
[10] H Aldweby and M Darus ldquoA subclass of harmonic univalentfunctions associated with 119902-analogue of Dziok-Srivastava oper-atorrdquo ISRN Mathematical Analysis vol 2013 Article ID 3823126 pages 2013
[11] H Aldweby and M Darus ldquoOn harmonic meromorphicfunctions associated with basic hypergeometric functionsrdquoTheScientific World Journal vol 2013 Article ID 164287 7 pages2013
[12] A Aral V Gupta and R P Agarwal Applications of q-Calculusin Operator Theory Springer New York NY USA 2013
[13] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[14] H Exton q-Hypergeometric Functions and Applications EllisHorwood Series Mathematics and Its Applications Ellis Hor-wood Chichester UK 1983
[15] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[16] M L Mogra ldquoApplications of Ruscheweyh derivatives andHadamard product to analytic functionsrdquo International Journalof Mathematics and Mathematical Sciences vol 22 no 4 pp795ndash805 1999
[17] K Inayat Noor and S Hussain ldquoOn certain analytic functionsassociated with Ruscheweyh derivatives and bounded Mocanuvariationrdquo Journal of Mathematical Analysis and Applicationsvol 340 no 2 pp 1145ndash1152 2008
[18] S L Shukla and V Kumar ldquoUnivalent functions defined byRuscheweyh derivativesrdquo International Journal of Mathematicsand Mathematical Sciences vol 6 no 3 pp 483ndash486 1983
[19] G S Rao and R Saravanan ldquoSome results concerning bestuniform coap- proximationrdquo Journal of Inequalities in Pure andApplied Math vol 3 no 2 p 24 2002
[20] G S Rao and K R Chandrasekaran ldquoCharacterization ofelements of best coapproximation in normed linear spacesrdquoPure and Applied Mathematical Sciences vol 26 pp 139ndash1471987
[21] M S Robertson ldquoCertain classes of starlike functionsrdquo TheMichiganMathematical Journal vol 32 no 2 pp 135ndash140 1985
6 Abstract and Applied Analysis
[22] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981
[23] S S Miller and P T Mocanu ldquoOn some classes of first-order differential subordinationsrdquo The Michigan MathematicalJournal vol 32 no 2 pp 185ndash195 1985
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
4 Abstract and Applied Analysis
Proof Following the same steps as in the proof ofTheorem 7and considering 119892(119911) = R120582
119902119891(119911)119911 the differential subordi-
nation (27) becomes
119892 (119911) +119902120582120572
[120582 + 1]119902
119911119863119902 (119892 (119911)) ≺1 + (2120573 minus 1) 119911
1 + 119911 (35)
Therefore
Re((R120582119902119891 (119911)
119911)
1119899
)
gt ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(1 + (1 minus 2120573) 119906
1 + 119906)119889119906)
1119899
= ([120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
times ((2120573 minus 1) +2 (1 minus 120573)
1 + 119906)119889119906)
1119899
= ( (2120573 minus 1) 119906[120582+1]119902119902
120582120572
+2 (1 minus 120573) [120582 + 1]119902
119902120582int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
1119899
(36)
Theorem 9 Let 120582 gt 0 and 0 le 120588 lt 1 Let 120574 be a complexnumber with 120574 = 0 and satisfy either |2120574(1 minus 120588)([120582 + 1]119902119902120582) minus1| le 1 or |2120574(1 minus 120588)([120582 + 1]119902119902120582) + 1| le 1 If 119891 isin A satisfiesthe condition
Re(R120582+1119902119891 (119911)
R120582119902119891 (119911)
) gt 120588 (119911 isin U) (37)
then
(
R120582119902119891 (119911)
119911)
120574
≺1
(1 minus 119911)2120574(1minus120588)([120582+1]119902119902
120582) (119911 isin U) (38)
where 119902(119911) is the best dominant
Proof Let
119901 (119911) = (
R120582119902119891 (119911)
119911)
120574
(119911 isin U) (39)
Then by making use of (11) (37) and (39) we obtain
1 +119902120582119911119863119902 (119901 (119911))
120574[120582 + 1]119902119901 (119911)≺1 + (1 minus 2120588) 119911
1 minus 119911 (119911 isin U) (40)
We now assume that
119902 (119911) =1
(1 minus 119911)2120574(1minus120588)[120582+1]119902119902
120582 120579 (119908) = 1
120601 (119908) =119902120582
120574[120582 + 1]119902119908
(41)
then 119902(119911) is univalent by condition of the theorem andLemma 4 Further it is easy to show that 119902(119911) 120579(119908) and 120601(119908)satisfy the conditions of Lemma 6 Note that the function
119876 (119911) = 119911119863119902 (119902 (119911)) 120601 (119902 (119911)) =2 (1 minus 120588) 119911
1 minus 119911(42)
is univalent starlike in U and
ℎ (119911) = 120579 (119902 (119911)) + 119876 (119911) =1 + (1 minus 2120588) 119911
1 minus 119911 (43)
Combining (40) and Lemma 6 we get the assertion ofTheorem 9
Theorem 10 Let 120572 lt 1 120582 gt 0 and minus1 le 119861119894 lt 119860 119894 le 1 If eachof the functions 119891119894 isin A satisfies the following subordinationcondition
(1 minus 120572)
R120582119902119891119894 (119911)
119911+ 120572
R120582+1119902119891119894 (119911)
119911≺ ℎ (119860 119894 119861119894 119911)
(44)
then
(1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911≺ ℎ (1 minus 2120574 minus1 119911) (45)
where
Θ (119911) = R120582
119902(1198911 lowast 1198912) (119911) (46)
120574 = 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906)
(47)
Proof we define the function ℎ119894 by
ℎ119894 (119911) = (1 minus 120572)
R120582119902119891119894 (119911)
119911+ 120572
R120582+1119902119891119894 (119911)
119911
(119891119894 isin A 119894 = 1 2)
(48)
we have ℎ119894(119911) isin 119875(120573119894) where 120573119894 = (1 minus119860 119894)(1 minus 119861119894) (119894 = 1 2)By making use of (11) and (48) we obtain
R120582
119902119891119894 (119911) =
[120582 + 1]119902
119902120582120572int
1
0
119905([120582+1]119902119902
120582120572)minus1
ℎ119894 (119905) 119889119905 (119894 = 1 2)
(49)
which in the light of (46) can show that
R120582
119902Θ (119911) =
[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)int
1
0
119905([120582+1]119902119902
120582120572)minus1
ℎ0 (119905) 119889119905
(50)
Abstract and Applied Analysis 5
where for convenience
ℎ0 (119911) = (1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911
=[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)
times int
1
0
119905([120582+1]119902119902
120582120572)minus1
(ℎ1 lowast ℎ2) (119905) 119889119905
(51)
Note that by using Lemma 2we have (ℎ1lowastℎ2) isin 119875(1205733) where1205733 = 1 minus 2(1 minus 1205731)(1 minus 1205732)
Now with an application of Lemma 3 we have
Re (ℎ0 (119911))
=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1 Re ((ℎ1 lowast ℎ2) (119906119911)) 119889119906
ge[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906 |119911|) 119889119906
gt[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906)119889119906
= 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906) = 120574
(52)
which shows that the desired assertion of Theorem 10 holds
Conflict of Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Authorsrsquo Contribution
Huda Aldweby and Maslina Darus read and approved thefinal manuscript
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments made to improve this paper
References
[1] F H Jackson ldquoOn 119902-definite integralsrdquoTheQuarterly Journal ofPure and Applied Mathematics vol 41 pp 193ndash203 1910
[2] F H Jackson ldquoOn 119902-functions and a certain difference opera-torrdquo Transactions of the Royal Society of Edinburgh vol 46 pp253ndash281 1908
[3] A Aral and V Gupta ldquoOn 119902-Baskakov type operatorsrdquoDemon-stratio Mathematica vol 42 no 1 pp 109ndash122 2009
[4] A Aral and V Gupta ldquoGeneralized 119902-Baskakov operatorsrdquoMathematica Slovaca vol 61 no 4 pp 619ndash634 2011
[5] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the 119902-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[6] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of some singular integrals in the unit diskrdquo Journal ofInequalities and Applications Article ID 17231 19 pages 2006
[7] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of generalized singular integrals in the unit diskrdquoJournal of the Korean Mathematical Society vol 43 no 2 pp425ndash443 2006
[8] A Aral ldquoOn the generalized Picard and Gauss Weierstrasssingular integralsrdquo Journal of Computational Analysis andAppli-cations vol 8 no 3 pp 249ndash261 2006
[9] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquo Matematichki Vesnik vol65 no 4 pp 454ndash465 2013
[10] H Aldweby and M Darus ldquoA subclass of harmonic univalentfunctions associated with 119902-analogue of Dziok-Srivastava oper-atorrdquo ISRN Mathematical Analysis vol 2013 Article ID 3823126 pages 2013
[11] H Aldweby and M Darus ldquoOn harmonic meromorphicfunctions associated with basic hypergeometric functionsrdquoTheScientific World Journal vol 2013 Article ID 164287 7 pages2013
[12] A Aral V Gupta and R P Agarwal Applications of q-Calculusin Operator Theory Springer New York NY USA 2013
[13] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[14] H Exton q-Hypergeometric Functions and Applications EllisHorwood Series Mathematics and Its Applications Ellis Hor-wood Chichester UK 1983
[15] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[16] M L Mogra ldquoApplications of Ruscheweyh derivatives andHadamard product to analytic functionsrdquo International Journalof Mathematics and Mathematical Sciences vol 22 no 4 pp795ndash805 1999
[17] K Inayat Noor and S Hussain ldquoOn certain analytic functionsassociated with Ruscheweyh derivatives and bounded Mocanuvariationrdquo Journal of Mathematical Analysis and Applicationsvol 340 no 2 pp 1145ndash1152 2008
[18] S L Shukla and V Kumar ldquoUnivalent functions defined byRuscheweyh derivativesrdquo International Journal of Mathematicsand Mathematical Sciences vol 6 no 3 pp 483ndash486 1983
[19] G S Rao and R Saravanan ldquoSome results concerning bestuniform coap- proximationrdquo Journal of Inequalities in Pure andApplied Math vol 3 no 2 p 24 2002
[20] G S Rao and K R Chandrasekaran ldquoCharacterization ofelements of best coapproximation in normed linear spacesrdquoPure and Applied Mathematical Sciences vol 26 pp 139ndash1471987
[21] M S Robertson ldquoCertain classes of starlike functionsrdquo TheMichiganMathematical Journal vol 32 no 2 pp 135ndash140 1985
6 Abstract and Applied Analysis
[22] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981
[23] S S Miller and P T Mocanu ldquoOn some classes of first-order differential subordinationsrdquo The Michigan MathematicalJournal vol 32 no 2 pp 185ndash195 1985
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
Abstract and Applied Analysis 5
where for convenience
ℎ0 (119911) = (1 minus 120572)
R120582119902Θ (119911)
119911+ 120572
R120582+1119902Θ (119911)
119911
=[120582 + 1]119902
1199021205821205721199111minus([120582+1]119902119902
120582120572)
times int
1
0
119905([120582+1]119902119902
120582120572)minus1
(ℎ1 lowast ℎ2) (119905) 119889119905
(51)
Note that by using Lemma 2we have (ℎ1lowastℎ2) isin 119875(1205733) where1205733 = 1 minus 2(1 minus 1205731)(1 minus 1205732)
Now with an application of Lemma 3 we have
Re (ℎ0 (119911))
=[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1 Re ((ℎ1 lowast ℎ2) (119906119911)) 119889119906
ge[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906 |119911|) 119889119906
gt[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
(21205733 minus 1 +2 (1 minus 1205733)
1 + 119906)119889119906
= 1 minus4 (1198601 minus 1198611) (1198602 minus 1198612)
(1 minus 1198611) (1 minus 1198612)
times (1 minus[120582 + 1]119902
119902120582120572int
1
0
119906([120582+1]119902119902
120582120572)minus1
1 + 119906119889119906) = 120574
(52)
which shows that the desired assertion of Theorem 10 holds
Conflict of Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Authorsrsquo Contribution
Huda Aldweby and Maslina Darus read and approved thefinal manuscript
Acknowledgments
Thework presented here was partially supported by AP-2013-009 and DIP-2013-001 The authors also would like to thankthe referees for the comments made to improve this paper
References
[1] F H Jackson ldquoOn 119902-definite integralsrdquoTheQuarterly Journal ofPure and Applied Mathematics vol 41 pp 193ndash203 1910
[2] F H Jackson ldquoOn 119902-functions and a certain difference opera-torrdquo Transactions of the Royal Society of Edinburgh vol 46 pp253ndash281 1908
[3] A Aral and V Gupta ldquoOn 119902-Baskakov type operatorsrdquoDemon-stratio Mathematica vol 42 no 1 pp 109ndash122 2009
[4] A Aral and V Gupta ldquoGeneralized 119902-Baskakov operatorsrdquoMathematica Slovaca vol 61 no 4 pp 619ndash634 2011
[5] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the 119902-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010
[6] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of some singular integrals in the unit diskrdquo Journal ofInequalities and Applications Article ID 17231 19 pages 2006
[7] G A Anastassiou and S G Gal ldquoGeometric and approximationproperties of generalized singular integrals in the unit diskrdquoJournal of the Korean Mathematical Society vol 43 no 2 pp425ndash443 2006
[8] A Aral ldquoOn the generalized Picard and Gauss Weierstrasssingular integralsrdquo Journal of Computational Analysis andAppli-cations vol 8 no 3 pp 249ndash261 2006
[9] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquo Matematichki Vesnik vol65 no 4 pp 454ndash465 2013
[10] H Aldweby and M Darus ldquoA subclass of harmonic univalentfunctions associated with 119902-analogue of Dziok-Srivastava oper-atorrdquo ISRN Mathematical Analysis vol 2013 Article ID 3823126 pages 2013
[11] H Aldweby and M Darus ldquoOn harmonic meromorphicfunctions associated with basic hypergeometric functionsrdquoTheScientific World Journal vol 2013 Article ID 164287 7 pages2013
[12] A Aral V Gupta and R P Agarwal Applications of q-Calculusin Operator Theory Springer New York NY USA 2013
[13] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[14] H Exton q-Hypergeometric Functions and Applications EllisHorwood Series Mathematics and Its Applications Ellis Hor-wood Chichester UK 1983
[15] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[16] M L Mogra ldquoApplications of Ruscheweyh derivatives andHadamard product to analytic functionsrdquo International Journalof Mathematics and Mathematical Sciences vol 22 no 4 pp795ndash805 1999
[17] K Inayat Noor and S Hussain ldquoOn certain analytic functionsassociated with Ruscheweyh derivatives and bounded Mocanuvariationrdquo Journal of Mathematical Analysis and Applicationsvol 340 no 2 pp 1145ndash1152 2008
[18] S L Shukla and V Kumar ldquoUnivalent functions defined byRuscheweyh derivativesrdquo International Journal of Mathematicsand Mathematical Sciences vol 6 no 3 pp 483ndash486 1983
[19] G S Rao and R Saravanan ldquoSome results concerning bestuniform coap- proximationrdquo Journal of Inequalities in Pure andApplied Math vol 3 no 2 p 24 2002
[20] G S Rao and K R Chandrasekaran ldquoCharacterization ofelements of best coapproximation in normed linear spacesrdquoPure and Applied Mathematical Sciences vol 26 pp 139ndash1471987
[21] M S Robertson ldquoCertain classes of starlike functionsrdquo TheMichiganMathematical Journal vol 32 no 2 pp 135ndash140 1985
6 Abstract and Applied Analysis
[22] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981
[23] S S Miller and P T Mocanu ldquoOn some classes of first-order differential subordinationsrdquo The Michigan MathematicalJournal vol 32 no 2 pp 185ndash195 1985
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 Abstract and Applied Analysis
[22] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981
[23] S S Miller and P T Mocanu ldquoOn some classes of first-order differential subordinationsrdquo The Michigan MathematicalJournal vol 32 no 2 pp 185ndash195 1985
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
