Inclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator

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1 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 R E S E A R C H Open Access Inclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator Nak Eun Cho 1 Min Yoon 2* * Correspondence: myoon@pknu.ac.kr 2 Department of Statistics, Pukyong National University, 45 Yongso-ro, Busan, Korea Full list of author information is available at the end of the article Abstract The purpose of the present paper is to investigate some inclusion properties of certain classes of analytic functions associated with a family of linear operators which are defined by means of the Hadamard product or convolution. Some invariant properties under convolution are also considered for the classes presented here. The results presented here include several previous known results as their special cases. MSC: 30C45 Keywords: subordination; Hadamard product or convolution; starlike function; convex function; close-to-convex function; linear operator; Choi-Saigo-Srivastava operator; Carlson-Shaffer operator; Ruscheweyh derivative operator 1 Introduction Let A denote the class of functions of the form f z=z + a k z k k=2 which are analytic in the open unit disk U = {z C : z <1}.Iff g are analytic in U,we say that f is subordinate to g,writtenf g or f z gz, if there exists an analytic function w in U with w0 = 0 ωz <1forz U such that f z =gωz.wedenotebys *, K C the subclasses of A consisting of all analytic functions which are, respectively, starlike, convex close-to-convex in U see, e.g.,srivastavaowa[1]. Let N be a class of all functions φ which are analytic univalent in U for which φuisconvexwithφ0 = 1 Re{φz} >0forz U. Making use of the principle of subordination between analytic functions, many authors investigated the subclasses S * η; φ, Kη; φcη, β; φ, ψoftheclassafor 0 η, β <1, φ, ψ N cf. [2][3], which are defined by { S * η; φ:= f A : { Kη; φ:= f A : 1 zf z 1η f z 1 1η 1+ zf z f z η } η φzinu, } φzinu 2013 Cho Yoon; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited.

2 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 Page 2 of 10 { Cη, β; φ, ψ:= f A : g S * 1 zf } z η; φs.t. 1β gz β ψzinu. For φz=ψz=1+z/1 z in the definitions above, we have the well-known classes S *, K C, respectively. Furthermore, for the function classes S * [A, B] K[A, B] investigated by Janowski [4], alsosee [5], it iseasily seen that S * 0; 1+Az = S * [A, B] 1 B < A 1 1+Bz K 0; 1+Az = K[A, B] 1 B < A 1. 1+Bz We now define the function φa, c; zby φa, c; z:= k=0 a k c k z k+1 z U; a R; c R\Z 0 ; Z 0 := {1,2,...}, 1.1 where ν k is the Pochhammer symbol or the shifted factorial defined in terms of the gamma function by ν k := Ɣν + k Ɣν 1 ifk =0ν C\{0}, = νν +1 ν + k 1 ifk N := {1,2,...} ν C. We also denote by La, c:a A the operator defined by La, cf z=φa, c; z f z z Uz U; f A, 1.2 where the symbol sts for the Hadamardproduct or convolution. Then it iseasily observed from definitions that L2, 1f z=zf z Ln +1,1f z=d n f z n > 1, where the symbol D n denotes the familiar Ruscheweyh derivative [6] also,see[7] for n N 0 := N {0}. The operator La, c, introduced studied by Carlson-Shaffer [8], has been used widely on the space of analytic univalent functions in U see also [9]. Corresponding to the function φa, c; zdefinedby1.1, we also introduce a function φ λ a, c; z given by φa, c; z φ λ a, c; z= z 1 z λ λ > Analogous to La, c, we now define the linear operator I λ a, cona as follows: I λ a, cf z=φ λ a, c; z f z a, c R\Z 0 ; λ >0;z U; f A. 1.4

3 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 Page 3 of 10 It is easily verified from definition 1.4that I 2 2, 1f z=f z I 2 1, 1f z=zf z. In particular, the operator I λ μ +1,1λ >0;μ >1wasintroducedbyChoiet al. [2] who investigated among other things several inclusion properties involving various subclasses of analytic univalent functions. For μ = n n N 0 λ =2,wealsonotethat the Choi-Saigo-Srivastava operator I λ μ +1,1f is the Noor integral operator of nth order of f studied by Liu [10]Nooret al. [1115]. By using the operator I λ a, c, we introduce the following classes of analytic functions for φ, ψ N, a, c R\Z 0, λ >00 η, β <1: S λ a,c η; φ:={ f A : I λ a, cf z S * η; φ }, K λ a,c η; φ:={ f A : I λ a, cf z K * η; φ }, C λ a,c η, β; φ, ψ:={ f A : I λ a, cf z C * η, β; φ, ψ }. We also note that f z Ka,c λ η; φ zf z Sa,c λ η; φ. 1.5 In particular, we set Sa,c λ η; 1+Az = Sa,c λ [η; A, B] 0 η <1;1 B < A 1 1+Bz K λ a,c η; 1+Az = Ka,c λ [η; A, B] 0 η <1,1 B < A 1. 1+Bz Recently, Sokoł [16] extended the results given by Choi et al. [2] making use of some interesting proof techniques due to Ruscheweyh [17] Ruscheweyh Sheil-Small [18]. In this paper, we investigate several inclusion properties of the classes Sa,c λ η; φ, Kλ a,c η; φ Ca,c λ η, β; φ, ψ. The integral-preserving properties in connection with the operator I λ a, c are also considered. Furthermore, relevant connections of the results presented here with those obtained in earlier works are pointed out. 2 Inclusion properties involving the operator I λ a, c The following lemmas will be required in our investigation. Lemma 2.1 Let φ λi a, c; z, φ λ a i, c; z φ λ a, c i ; zi =1,2be defined by 1.3. Then, for λ i >0,a i, c i R\Z 0 i =1,2, φ λ1 a, c; z=φ λ2 a, c; z f λ1,λ 2 z, 2.1 φ λ a 2, c; z=φ λ a 1, c; z f a1,a 2 z, 2.2

4 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 Page 4 of 10 φ λ a, c 1 ; z=φ λ a, c 2 ; z f c1,c 2 z, 2.3 where f s,t z= k=0 s k t k z k+1 z U. 2.4 Proof From definition 1.3, we know that φ λ a, c; z= k=0 c k λ k a k 1 k z k+1 z U. 2.5 Therefore, equations 2.1, followfrom 2.5 immediately. Lemma 2.2 [17, pp.60-61] Let t s >0.If t 2 or s + t 3, then the function defined by 2.4 belongs to the class K. Lemma 2.3 [18] Let f K g S *. Then, for every analytic function h in U, f hgu f gu cohu, where cohu denotes the closed convex hull of hu. At first, the inclusion relationship involving the class Sa,c λ η; φ is contained in Theorem 2.1 below. Theorem 2.1 Let λ 2 λ 1 >0,a, c R\Z 0,0 η <1 φ N. If λ 2 2 or λ 1 + λ 2 3, then S λ 2 a,c η; φ Sλ 1 a,c η; φ. Proof Let f S λ 2 a,cη; φ. From the definition of S λ 2 a,cη; φ, we have 1 ziλ2 a, cf z η = φ ωz z U, 2.6 1η I λ2 a, cf z where w is analytic in U with ωz <1z Uw0 = 0. By using 1.4, , we get zi λ1 a, cf z I λ1 a, cf z = zφ λ 1 a, c; z f z φ λ1 a, c; z f z = f λ 1,λ 2 z zi λ2 a, cf z f λ1,λ 2 z I λ2 a, cf z = zφ λ 2 a, c; z f λ1,λ 2 z f z φ λ2 a, c; z f λ1,λ 2 z f z = f λ 1,λ 2 z [1 ηφωz + η]i λ2 a, cf z. 2.7 f λ1,λ 2 z I λ2 a, cf z

5 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 Page 5 of 10 It follows from 2.6 Lemma 2.2 that I λ2 a, cf z S * f λ1,λ 2 K,respectively.Let usput sωz := 1 ηφωz+η. Then, by applying Lemma 2.3to 2.7, we obtain {f λ1,λ 2 [1 ηφw+η]i λ2 a, cf } U cos wu, 2.8 {f λ1,λ 2 I λ2 a, cf } since s is convex univalent. Therefore, from the definition of subordination 2.8, we have 1 ziλ1 a, cf z η φz z U, 1η I λ1 a, cf z or, equivalently, f S λ 1 a,cη; φ, which completes the proof of Theorem 2.1. By using equations 1.4, , we have the following theorems. Theorem 2.2 Let λ >0,a 2 a 1 >0,c R\Z 0,0 η <1 φ N. If a 2 2 or a 1 + a 2 3, then Sa λ 1,c η; φ Sλ a 2,cη; φ. Theorem 2.3 Let λ >0,a R\Z 0, c 2 c 1 >0,0 η <1 φ N. If c 2 2 or c 1 + c 2 3, then S λ a,c 2 η; φ S λ η; φ. Next, we prove the inclusion theorem involving the class Ka,c λ η; φ. Theorem 2.4 Let λ 2 λ 1 >0,a, c R\Z 0,0 η <1 φ N. If λ 2 2 or λ 1 + λ 2 3, then K λ 2 a,c η; φ Kλ 1 a,c η; φ. Proof Applying 1.5Theorem2.1,weobservethat f z K λ 2 a,c η; φ I λ 2 a, cf z Kη; φ z I λ2 a, cf z S * η; φ I λ2 a, c zf z S * η; φ zf z S λ 2 a,c η; φ zf z S λ 1 a,c η; φ I λ1 a, c zf z S * η; φ z I λ1 a, cf z S * η; φ I λ1 a, cf z Kη; φ f z K λ 1 a,c η; φ, which evidently proves Theorem 2.4.

6 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 Page 6 of 10 By using a similar method as in the proof of Theorem 2.4, we obtain the following two theorems. Theorem 2.5 Let λ >0,a 2 a 1 >0,c R\Z 0,0 η <1 φ N. If a 2 2 or a 1 + a 2 3, then Ka λ 1,c η; φ Kλ a 2,cη; φ. Theorem 2.6 Let λ >0,a R\Z 0, c 2 c 1 >0,0 η <1 φ N. If c 2 2 or c 1 + c 2 3, then K λ a,c 2 η; φ K λ η; φ. Taking φz=1+az/1 + Bz1 B < A 1; z UinTheorems ,wehavethe following corollaries below. Corollary 2.1 Let λ 2 λ 1 >0 let λ 2 min{2, 3 λ 1 }, a 2 a 1 >0 a 2 min{2, 3 a 1 }. Then S λ 2 a 1,c [η; A, B] Sλ 1 a 1,c [η; A, B] Sλ 1 a 2,c [η; A, B] c R\Z 0 ;0 η <1;1 B < A 1, K λ 2 a 1,c [η; A, B] Kλ 1 a 1,c [η; A, B] Kλ 1 a 2,c [η; A, B] c R\Z 0 ;0 η <1;1 B < A 1. Corollary 2.2 Let a 2 a 1 >0 let a 2 min{2, 3 a 1 }, c 2 c 1 >0 c 2 min{2, 3 c 1 }. Then S λ a 1,c 2 [η; A, B] S λ a 1,c 1 [η; A, B] S λ a 2,c 1 [η; A, B] λ >0;0 η <1;1 B < A 1, K λ a 1,c 2 [η; A, B] K λ a 1,c 1 [η; A, B] K λ a 2,c 1 [η; A, B] λ >0;0 η <1;1 B < A 1. Corollary 2.3 Let λ 2 λ 1 >0 let λ 2 min{2, 3 λ 1 }, c 2 c 1 >0 c 2 min{2, 3 c 1 }. Then S λ 2 a,c 2 [η; A, B] S λ 2 [η; A, B] S λ 1 [η; A, B] a R\Z 0 ;0 η <1;1 B < A 1, K λ 2 a,c 2 [η; A, B] K λ 2 [η; A, B] K λ 1 [η; A, B] a R\Z 0 ;0 η <1;1 B < A 1. To prove the theorems below, we need the following lemma. Lemma 2.4 [16] Let φ N. If f K q S * η; φ, then f q S * η; φ.

7 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 Page 7 of 10 Proof Let q S * η; φ. Then zq z= [ 1 ηφ ωz + η ] qz z U, where ω is an analytic function in U with ωz <1z Uw0 = 0. Thus we have zf z qz f z qz = f z zq z f z qz = f z [1 ηφωz + η]qz f z qz z U. 2.9 By using similar arguments to those used in the proof of Theorem 2.1,weconcludethat 2.9 is subordinated to φ in U so f q S * η; φ. Finally, we give the inclusion properties involving the class Ca,c λ η, β; φ, ψ. Theorem 2.7 Let λ 2 λ 1 >0 λ 2 min{2, 3 λ 1 }, let a 2 a 1 >0 a 2 min{2, 3 a 1 }. Then C λ 2 a 1,c η, β; φ, ψ Cλ 1 a 1,c η, β; φ, ψ Cλ 1 a 2,cη, β; φ, ψ c R\Z 0 ;0 η, β <1;φ, ψ N. Proof We begin by proving that C λ 2 a 1,c η, β; φ, ψ Cλ 1 a 1,cη, β; φ, ψ. Let f C λ 2 a 1,cη, β; φ, ψ. Then there exists a function q 2 S * η; φsuchthat 1 ziλ2 a 1, cf z β ψz z U β q 2 z From 2.10, we obtain z I λ2 a 1, cf z = [ 1 βψ ωz + β ] q2 z z U, where w is an analytic function in U with ωz <1z Uw0 = 0. By virtue of 2.3, Lemma 2.2 Lemma 2.4, weseethatf λ1,λ 2 z q 2 z q 1 z belongstos * η; φ. Then, making use of 2.1, we have 1 ziλ1 a 1, cf z β = 1 fλ1,λ 2 z zi λ2 a 1, cf z β 1β q 1 z 1β f λ1,λ 2 z q 2 z = 1 fλ1,λ 2 z [1 βψωz + β]q 2 z 1β f λ1,λ 2 z q 2 z β ψz z U. Therefore we prove that f C λ 1 a 1,cη, β; φ, ψ. For the second part, by using arguments similar to those detailed above with 2.2, we obtain C λ 1 a 1,c η, β; φ, ψ Cλ 1 a 2,cη, β; φ, ψ. Thus the proof of Theorem 2.7 is completed.

8 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 Page 8 of 10 The following results can be obtained by using the same techniques as in the proof of Theorem 2.7, so we omitthedetailed proofsinvolved. Theorem 2.8 Let a 2 a 1 >0 a 2 min{2, 3 a 1 }, let c 2 c 1 >0 c 2 min{2, 3 c 1 }. Then C λ a 1,c 2 η, β; φ, ψ C λ a 1,c 1 η, β; φ, ψ C λ a 2,c 1 η, β; φ, ψ λ >0;0 η, β <1;φ, ψ N. Theorem 2.9 Let λ 2 λ 1 >0 λ 2 min{2, 3 λ 1 }, let c 2 c 1 >0 c 2 min{2, 3 c 1 }. Then C λ 2 a,c 2 η, β; φ, ψ C λ 2 η, β; φ, ψ C λ 1 η, β; φ, ψ a R\Z 0 ;0 η, β <1;φ, ψ N. Remark 2.1 i Taking λ 2 = λ 1 +1λ 1 1, a 2 = a 1 +1a 1 1, c =1η = β =0in Theorems , Theorems Theorem 2.7, wehavetheresultsobtainedby Choi et al. [2], which extend the results earlier given by Noor et al. [12, 14]Liu[10]. ii For a = μ +1μ >1,c =1η = β =0,Theorems ,Theorems Theorem 2.7 reduce the corresponding results obtained by Sokoł [16]. 3 Inclusion properties involving various operators The next theorem shows that the classes Sa,c λ η; φ, Kλ a,c η; φcλ a,c η, β; φ, ψareinvariant under convolution with convex functions. Theorem 3.1 Let λ >0,a >0,c R\Z 0,0 η, β <1,φ, ψ N let g K. Then i f Sa,c λ η; φ= g f Sλ a,c η; φ, ii f Ka,c λ η; φ= g f Kλ a,c η; φ, iii f Ca,c λ η, β; φ, ψ= g f Cλ a,c η, β; φ, ψ. Proof i Let f Sa,c λ η; φ. Then we have 1 ziλ a, cg f z η = 1 gz ziλ a, cf z η. 1η I λ a, cg f z 1η gz I λ a, cf z By using the same techniques as in the proof of Theorem 2.1,weobtaini. ii Let f Ka,c λ η; φ. Then, by 1.5, zf z Sa,c λ η; φ hence from i, gz zf z Sa,c λ η; φ. Since gz zf z=zg f z, we have ii applying 1.5onceagain. iii Let f Ca,c λ η, β; φ, ψ. Then there exists a function q S* η; φsuchthat z I λ a, cf z = [ 1 βψ ωz + β ] qz z U,

9 Cho Yoon Journal of Inequalities Applications 2013, 2013:83 Page 9 of 10 where w is an analytic function in U with ωz <1z Uw0 = 0. From Lemma 2.4, we have that g q S * η; φ. Since 1 ziλ a, cg f z β = 1 gz ziλ a, cf z β 1β g qz 1β gz qz = 1 gz [1 βψωz + β]qz 1β gz qz β ψz z U, we obtain iii. Now we consider the following operators defined by 1 z= k=1 1+c k + c zk Re{c} 0; z U z= 1 [ ] 1xz log 1x log 1=0; x 1, x 1;z U z It is well known [19], see also [6, 20] that the operators 1 2 are convex univalent in U. Therefore we have the following result, which can be obtained from Theorem 3.1 immediately. Corollary 3.1 Let a >0,λ >0,c R\Z 0,0 η, β <1,φ, ψ N let i i =1,2be defined by Then i f Sa,c λ η; φ= i f Sa,c λ η; φ, ii f Ka,c λ η; φ= i f Ka,c λ η; φ, iii f Ca,c λ η, β; φ, ψ= i f Ca,c λ η, β; φ, ψ. Remark 3.1 Letting a = μ +1μ >1,c =1η = β =0inTheorem3.1, wehavethe corresponding results given by Sokoł [16]. Competing interests The authors declare that they have no competing interests. Authors contributions All authors jointly worked on the results they read approved the final manuscript. Author details 1 Department of Applied Mathematics, Pukyong National University, 45 Yongso-ro, Busan, Korea. 2 Department of Statistics, Pukyong National University, 45 Yongso-ro, Busan, Korea. Acknowledgements Dedicated to Professor Hari M Srivastava. The authors would like to express their thanks to the referees for some valuable comments regarding a previous version of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science Technology No Received: 27 November 2012 Accepted: 4 February 2013 Published: 1 March 2013

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