Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 931230, 18 pages doi:10.1155/2009/931230 Research Article Some Subclasses of Meromorphic Functions Associated with a Family of Integral Operators Zhi-Gang Wang,1 Zhi-Hong Liu,2 and Yong Sun3 School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha, Hunan 410114, China School of Mathematics, Honghe University, Mengzi, Yunnan 661100, China Department of Mathematics, Huaihua University, Huaihua, Hunan 418008, China Correspondence should be addressed to Zhi-Gang Wang, zhigangwang@foxmail.com Received 11 July 2009; Accepted September 2009 Recommended by Narendra Kumar Govil Making use of the principle of subordination between analytic functions and a family of integral operators defined on the space of meromorphic functions, we introduce and investigate some new subclasses of meromorphic functions Such results as inclusion relationships and integralpreserving properties associated with these subclasses are proved Several subordination and superordination results involving this family of integral operators are also derived Copyright q 2009 Zhi-Gang Wang et al This 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 Introduction and Preliminaries Let Σ denote the class of functions of the form f z z ∞ ak zk , 1.1 k which are analytic in the punctured open unit disk U∗ : {z : z ∈ C, < |z| < 1} : U \ {0} 1.2 Let f, g ∈ Σ, where f is given by 1.1 and g is defined by g z z ∞ k bk zk 1.3 Journal of Inequalities and Applications Then the Hadamard product or convolution f ∗ g of the functions f and g is defined by f ∗g z : ∞ z ak bk zk : g ∗ f z 1.4 k Let P denote the class of functions of the form ∞ p z pk zk , 1.5 k which are analytic and convex in U and satisfy the condition R p z >0 z∈U 1.6 For two functions f and g, analytic in U, we say that the function f is subordinate to g in U, and write f z ≺g z , 1.7 if there exists a Schwarz function ω, which is analytic in U with ω |ω z | < 0, z∈U 1.8 such that f z g ω z z∈U 1.9 Indeed, it is known that f z ≺g z ⇒f g , f U ⊂g U 1.10 Furthermore, if the function g is univalent in U, then we have the following equivalence: f z ≺ g z ⇐⇒ f g 0, f U ⊂g U 1.11 Analogous to the integral operator defined by Jung et al , Lashin introduced and investigated the following integral operator: Qα,β : Σ −→ Σ 1.12 Journal of Inequalities and Applications defined, in terms of the familiar Gamma function, by Qα,β f z Γ β α z Γ β Γ α zβ tβ − t z α−1 Γ k β f t dt 1.13 Γ β z ∞ α Γ β k 1Γ k β α Γ k β ak zk α α > 0; β > 0; z ∈ U∗ By setting ∞ Γ β z fα,β z : Γ β α Γ k k β zk α > 0; β > 0; z ∈ U∗ , 1.14 λ we define a new function fα,β z in terms of the Hadamard product or convolution : λ fα,β z ∗ fα,β z z 1−z λ α > 0; β > 0; λ > 0; z ∈ U∗ 1.15 Then, motivated essentially by the operator Qα,β , we now introduce the operator Qλ : Σ −→ Σ, α,β 1.16 which is defined as λ Qλ f z : fα,β z ∗ f z α,β z ∈ U∗ ; f ∈ Σ , 1.17 where and throughout this paper unless otherwise mentioned the parameters α, β, and λ are constrained as follows: α > 0; β > 0; λ > 1.18 We can easily find from 1.14 , 1.15 , and 1.17 that z Qλ f z α,β where λ k Γ β Γ β α ∞ k λ k Γ k β ak zk k !Γ k β α z ∈ U∗ , 1.19 is the Pochhammer symbol defined by λ k : ⎧ ⎨1, ⎩λ λ Clearly, we know that Q1 α,β k ··· λ Qα,β k−1 , 0, k ∈ N : {1, 2, } 1.20 Journal of Inequalities and Applications It is readily verified from 1.19 that z Qλ f α,β z Qλ α 1,β f z z β λQλ f z − λ α,β Qλ f z , α,β α Qλ f z − β α,β Qλ α α 1.21 1,β f 1.22 z By making use of the principle of subordination between analytic functions, we introduce the subclasses MS∗ η; φ , MK η; φ , MC η, δ; φ, ψ , and MQC η, δ; φ, ψ of the class Σ which are defined by MS∗ η; φ : f ∈Σ: zf z − −η 1−η f z MK η; φ : f ∈Σ: zf z −1 − −η 1−η f z MC η, δ; φ, ψ : η, δ < 1; z ∈ U η, δ < 1; z ∈ U η < 1; z ∈ U φ ∈ P; , η < 1; z ∈ U zf z − −δ 1−δ g z , ≺ψ z , f ∈ Σ : ∃ g ∈ MK η; φ such that φ, ψ ∈ P; φ ∈ P; ≺φ z f ∈ Σ : ∃ g ∈ MS∗ η; φ such that φ, ψ ∈ P; MQC η, δ; φ, ψ : ≺φ z 1−δ − zf z g z −δ ≺ψ z 1.23 Indeed, the above mentioned function classes are generalizations of the general meromorphic starlike, meromorphic convex, meromorphic close-to-convex and meromorphic quasi-convex functions in analytic function theory see, for details, 3–12 Next, by using the operator defined by 1.19 , we define the following subclasses λ MSα,β η; φ , MKλ η; φ , MCλ η, δ; φ, ψ , and MQCλ η, δ; φ, ψ of the class Σ: α,β α,β α,β λ MSα,β η; φ : f ∈ Σ : Qλ f ∈ MS∗ η; φ α,β , MKλ η; φ : α,β f ∈ Σ : Qλ f ∈ MK η; φ α,β , 1.24 MCλ η, δ; φ, ψ : α,β MQCλ η, δ; φ, ψ : α,β f ∈ Σ : Qλ f ∈ MC η, δ; φ, ψ α,β f ∈ Σ : Qλ f ∈ MQC η, δ; φ, ψ α,β , Journal of Inequalities and Applications Obviously, we know that λ f ∈ MKλ η; φ ⇐⇒ −zf ∈ MSα,β η; φ , α,β 1.25 f ∈ MQCλ η, δ; φ, ψ ⇐⇒ −zf ∈ MCλ η, δ; φ, ψ α,β α,β 1.26 In order to prove our main results, we need the following lemmas Lemma 1.1 see 13 Let κ, ϑ ∈ C Suppose also that m is convex and univalent in U with m0 If u is analytic in U with u R κm z 1, ϑ >0 z∈U 1.27 1, then the subordination uz zu z ≺m z κu z ϑ 1.28 implies that u z ≺m z 1.29 Lemma 1.2 see 14 Let h be convex univalent in U and let ζ be analytic in U with Rζ z If q is analytic in U and q 0 z∈U 1.30 h , then the subordination ζ z zq z ≺ h z 1.31 q z ≺h z q z 1.32 implies that The main purpose of the present paper is to investigate some inclusion relationships and integral-preserving properties of the subclasses λ MSα,β η; φ , MKλ η; φ , α,β MCλ η, δ; φ, ψ , α,β MQCλ η, δ; φ, ψ α,β 1.33 of meromorphic functions involving the operator Qλ Several subordination and superordiα,β nation results involving this operator are also derived 6 Journal of Inequalities and Applications The Main Inclusion Relationships We begin by presenting our first inclusion relationship given by Theorem 2.1 Theorem 2.1 Let η < and φ ∈ P with max R φ z α−η 1−η λ−η β , 1−η < z∈U z∈U 2.1 Then λ λ λ MSα,β1 η; φ ⊂ MSα,β η; φ ⊂ MSα 1,β η; φ 2.2 Proof We first prove that λ λ MSα,β1 η; φ ⊂ MSα,β η; φ 2.3 λ Let f ∈ MSα,β1 η; φ and suppose that ⎛ hz : where h is analytic in U with h λ ⎜ ⎝− 1−η ⎞ z Qλ f α,β z Qλ f z α,β ⎟ − η⎠, 2.4 Combining 1.21 and 2.4 , we find that Qλ f z α,β − 1−η h z −η Qλ f z α,β λ 2.5 Taking the logarithmical differentiation on both sides of 2.5 and multiplying the resulting equation by z, we get ⎛ ⎜ ⎝− 1−η z Qλ f α,β ⎞ z Qλ f z α,β ⎟ − η⎠ h z zh z − 1−η h z −η λ ≺φ z 2.6 λ By virtue of 2.1 , an application of Lemma 1.1 to 2.6 yields h ≺ φ, that is f ∈ MSα,β η; φ Thus, the assertion 2.3 of Theorem 2.1 holds λ To prove the second part of Theorem 2.1, we assume that f ∈ MSα,β η; φ and set ⎛ gz : ⎜ ⎝− 1−η z Qλ α ⎞ f 1,β Qλ 1,β f α z z ⎟ − η⎠, 2.7 Journal of Inequalities and Applications where g is analytic in U with g Combining 1.22 , 2.1 , and 2.7 and applying the λ similar method of proof of the first part, we get g ≺ φ, that is f ∈ MSα 1,β η; φ Therefore, the second part of Theorem 2.1 also holds The proof of Theorem 2.1 is evidently completed Theorem 2.2 Let η < and φ ∈ P with 2.1 holds Then MKλ η; φ ⊂ MKλ η; φ ⊂ MKλ α,β α,β α 1,β η; φ 2.8 Proof In view of 1.25 and Theorem 2.1, we find that f ∈ MKλ η; φ ⇐⇒ Qλ f ∈ MK η; φ α,β α,β ⇐⇒ −z Qλ f α,β ∈ MS∗ η; φ ⇐⇒ Qλ −zf α,β ∈ MS∗ η; φ λ ⇐⇒ −zf ∈ MSα,β1 η; φ λ ⇒ −zf ∈ MSα,β η; φ 2.9 ∈ MS∗ η; φ ⇐⇒ Qλ −zf α,β ⇐⇒ −z Qλ f ∈ MS∗ η; φ α,β ⇐⇒ Qλ f ∈ MK η; φ α,β ⇐⇒ f ∈ MKλ η; φ , α,β λ f ∈ MKλ η; φ ⇐⇒ −zf ∈ MSα,β η; φ α,β λ ⇒ −zf ∈ MSα ⇐⇒ Qλ α 1,β ⇐⇒ Qλ α 1,β f 1,β −zf η; φ ∈ MS∗ η; φ 2.10 ∈ MK η; φ ⇐⇒ f ∈ MKλ α 1,β η; φ Combining 2.9 and 2.10 , we deduce that the assertion of Theorem 2.2 holds Theorem 2.3 Let η < 1, δ < and φ, ψ ∈ P with 2.1 holds Then MCλ η, δ; φ, ψ ⊂ MCλ η, δ; φ, ψ ⊂ MCλ α,β α,β α 1,β η, δ; φ, ψ 2.11 Journal of Inequalities and Applications Proof We begin by proving that MCλ η, δ; φ, ψ ⊂ MCλ η, δ; φ, ψ α,β α,β 2.12 Let f ∈ MCλ η, δ; φ, ψ Then, by definition, we know that α,β ⎛ ⎜ ⎝− 1−δ ⎞ z Qλ f α,β Qλ g α,β z z ⎟ − δ⎠ ≺ ψ z 2.13 λ λ with g ∈ MSα,β1 η; φ , Moreover, by Theorem 2.1, we know that g ∈ MSα,β η; φ , which implies that ⎛ qz : ⎜ ⎝− 1−η ⎞ z Qλ g α,β Qλ g α,β z z ⎟ − η⎠ ≺ φ z 2.14 We now suppose that ⎛ pz : where p is analytic in U with p − 1−δ p z ⎜ ⎝− 1−δ z Qλ f α,β Qλ g z α,β ⎞ z ⎟ − δ⎠, 2.15 Combining 1.21 and 2.15 , we find that δ Qλ g z α,β λQλ f z − λ α,β Qλ f z α,β 2.16 Differentiating both sides of 2.16 with respect to z and multiplying the resulting equation by z, we get − − δ zp z − 1−δ p z δ − 1−η q z −η λ λ z Qλ f α,β Qλ g z α,β z 2.17 In view of 1.21 , 2.14 , and 2.17 , we conclude that ⎛ ⎜ ⎝− 1−δ ⎞ z Qλ f α,β Qλ g α,β z z ⎟ − δ⎠ p z zp z − 1−η q z −η λ ≺ψ z 2.18 By noting that 2.1 holds and q z ≺φ z , 2.19 Journal of Inequalities and Applications we know that R − 1−η q z −η λ > 2.20 Thus, an application of Lemma 1.2 to 2.18 yields p z ≺ψ z , 2.21 that is f ∈ MCλ η, δ; φ, ψ , which implies that the assertion 2.12 of Theorem 2.3 holds α,β By virtue of 1.22 and 2.1 , making use of the similar arguments of the details above, we deduce that MCλ η, δ; φ, ψ ⊂ MCλ α,β α 1,β η, δ; φ, ψ 2.22 The proof of Theorem 2.3 is thus completed Theorem 2.4 Let η < 1, δ < and φ, ψ ∈ P with 2.1 holds Then MQCλ η, δ; φ, ψ ⊂ MQCλ η, δ; φ, ψ ⊂ MQCλ α,β α,β α 1,β η, δ; φ, ψ 2.23 Proof In view of 1.26 and Theorem 2.3, and by similarly applying the method of proof of Theorem 2.2, we conclude that the assertion of Theorem 2.4 holds A Set of Integral-Preserving Properties In this section, we derive some integral-preserving properties involving two families of integral operators λ Theorem 3.1 Let f ∈ MSα,β η; φ with φ ∈ P and R φ z < R ν −η 1−η z ∈ U; R ν > 3.1 Then the integral operator Fν f defined by ν−1 zν Fν f : Fν f z z tν−1 f t dt z ∈ U; R ν > 3.2 λ belongs to the class MSα,β η; φ λ Proof Let f ∈ MSα,β η; φ Then, from 3.2 , we find that z Qλ Fν f α,β z νQλ Fν f z α,β ν − Qλ f z α,β 3.3 10 Journal of Inequalities and Applications By setting ⎛ Pz : ⎜ ⎝− 1−η Qλ Fν α,β we observe that P is analytic in U with P − 1−η P z −η ⎞ z Qλ Fν f α,β z f z ⎟ − η⎠, 3.4 It follows from 3.3 and 3.4 that ν−1 ν Qλ f z α,β Qλ Fν f z α,β 3.5 Differentiating both sides of 3.5 with respect to z logarithmically and multiplying the resulting equation by z, we get ⎛ P z zP z − 1−η P z −η ν ⎜ ⎝− 1−η ⎞ z Qλ f α,β Qλ f α,β z z ⎟ − η⎠ ≺ φ z 3.6 Since 3.1 holds, an application of Lemma 1.1 to 3.6 yields ⎛ ⎜ ⎝− 1−η z Qλ Fν f α,β Qλ Fν α,β ⎞ z f z ⎟ − η⎠ ≺ φ z , 3.7 which implies that the assertion of Theorem 3.1 holds Theorem 3.2 Let f ∈ MKλ η; φ with φ ∈ P and 3.1 holds Then the integral operator Fν f α,β defined by 3.2 belongs to the class MKλ η; φ α,β Proof By virtue of 1.25 and Theorem 3.1, we easily find that λ f ∈ MKλ η; φ ⇐⇒ −zf ∈ MSα,β η; φ α,β ⇒ Fν −zf ⇐⇒ −z Fν f λ ∈ MSα,β η; φ ∈ MS∗ η; φ ⇐⇒ Fν f ∈ MKλ η; φ α,β The proof of Theorem 3.2is evidently completed 3.8 Journal of Inequalities and Applications 11 Theorem 3.3 Let f ∈ MCλ η, δ; φ, ψ with φ ∈ P and 3.1 holds Then the integral operator α,β Fν f defined by 3.2 belongs to the class MCλ η, δ; φ, ψ α,β Proof Let f ∈ MCλ η, δ; φ, ψ Then, by definition, we know that there exists a function α,β g ∈ MS∗ η; φ such that ⎛ ⎜ ⎝− 1−η ⎞ z Qλ f α,β Qλ g α,β z z ⎟ − η⎠ ≺ ψ z 3.9 Since g ∈ MS ∗ η; φ , by Theorem 3.1, we easily find that Fν g ∈ MS∗ η; φ , which implies that ⎛ ⎜ ⎝− 1−η Hz : ⎞ z Qλ Fν g α,β z Qλ Fν g z α,β ⎟ − η⎠ ≺ φ z 3.10 We now set ⎛ ⎜ ⎝− 1−δ Qz : where Q is analytic in U with Q − 1−δ Q z z Qλ Fν f α,β ⎞ z Qλ Fν g z α,β ⎟ − δ⎠, 3.11 From 3.3 , and 3.11 , we get δ Qλ Fν g z α,β νQλ Fν f z α,β ν − Qλ f z α,β 3.12 Combining 3.10 , 3.11 , and 3.12 , we find that − − δ zQ z − 1−δ Q z δ − 1−η H z −η ν ν−1 z Qλ f α,β z Qλ Fν g z α,β 3.13 By virtue of 1.21 , 3.10 , and 3.13 , we deduce that ⎛ ⎜ ⎝− 1−δ ⎞ z Qλ f α,β Qλ g α,β z z ⎟ − δ⎠ Qz zQ z − 1−η H z −η ν ≺ψ z 3.14 12 Journal of Inequalities and Applications The remainder of the proof of Theorem 3.3 is much akin to that of Theorem 2.3 We, therefore, choose to omit the analogous details involved We thus find that Q z ≺ψ z , 3.15 which implies that Fν f ∈ MCλ η, δ; φ, ψ The proof of Theorem 3.3 is thus completed α,β Theorem 3.4 Let f ∈ MQCλ η, δ; φ, ψ with φ ∈ P and 3.1 holds Then the integral operator α,β Fν f defined by 3.2 belongs to the class MQCλ η, δ; φ, ψ α,β Proof In view of 1.26 and Theorem 3.3, and by similarly applying the method of proof of Theorem 3.2, we deduce that the assertion of Theorem 3.4 holds λ Theorem 3.5 Let f ∈ MSα,β η; φ with φ ∈ P and R σ − ηξ − − η ξφ z z ∈ U; ξ / >0 3.16 σ Then the function Kξ f ∈ Σ defined by σ σ Qλ Kξ f : Qλ Kξ f z α,β α,β σ−ξ zσ z ξ tσ−1 Qλ f t α,β 1/ξ z ∈ U∗ ; ξ / dt 3.17 λ belongs to the class MSα,β η; φ λ Proof Let f ∈ MSα,β η; φ and suppose that ⎛ M z : ⎜ ⎝− 1−η ⎞ σ z Qλ Kξ f α,β z σ Qλ Kξ f z α,β ⎟ − η⎠ 3.18 Combining 3.17 and 3.18 , we have ⎛ σ − ηξ − − η ξM z σ−ξ ⎝ ⎞ξ Qλ f z α,β σ Qλ Kξ f z α,β ⎠ 3.19 Now, in view of 3.17 , 3.18 , and 3.19 , we get ⎛ M z zM z σ − ηξ − − η ξM z ⎜ ⎝− 1−η ⎞ z Qλ f α,β Qλ f α,β z z ⎟ − η⎠ ≺ φ z 3.20 Journal of Inequalities and Applications 13 Since 3.16 holds, an application of Lemma 1.1 to 3.20 yields M z ≺φ z , 3.21 σ λ that is, Kξ f ∈ MSα,β η; φ We thus complete the proof of Theorem 3.5 σ Theorem 3.6 Let f ∈ MKλ η; φ with φ ∈ P and 3.16 holds Then the function Kξ f ∈ Σ α,β defined by 3.17 belongs to the class MKλ η; φ α,β Proof By virtue of 1.25 and Theorem 3.5, and by similarly applying the method of proof of Theorem 3.2, we conclude that the assertion of Theorem 3.6 holds σ Theorem 3.7 Let f ∈ MCλ η, δ; φ, ψ with φ ∈ P and 3.16 holds Then the function Kξ f ∈ Σ α,β defined by 3.17 belongs to the class MCλ η, δ; φ, ψ α,β Proof Let f ∈ MCλ η, δ; φ, ψ Then, by definition, we know that there exists a function α,β g ∈ MS∗ η; φ such that 3.9 holds Since g ∈ MS∗ η; φ , by Theorem 3.5, we easily find that σ Kξ g ∈ MS∗ η; φ , which implies that Rz : ⎛ ⎞ σ λ z ⎜ z Qα,β Kξ g ⎟ − η⎠ ≺ φ z ⎝− σ 1−η Qλ Kξ g z α,β 3.22 We now set ⎛ ⎞ σ λ z ⎜ z Qα,β Kξ f ⎟ − δ⎠, ⎝− σ 1−δ Qλ Kξ g z α,β Dz : where D is analytic in U with D −ξ − δ D z 3.23 From 3.17 and 3.23 , we get σ δ Qλ Kξ g z α,β σ δQλ Kξ f z α,β δ − ξ Qλ f z α,β 3.24 Combining 3.22 , 3.23 , and 3.24 , we find that −ξ − δ zD z − 1−δ D z δ − − η ξR z − ηξ δ δ−ξ z Qλ f α,β z σ Qλ Kξ g z α,β 3.25 Furthermore, by virtue of 1.22 , 3.22 , and 3.25 , we deduce that ⎛ ⎜ ⎝− 1−δ ⎞ z Qλ f α,β Qλ g α,β z z ⎟ − δ⎠ D z zD z − − η ξR z − ηξ δ ≺ψ z 3.26 14 Journal of Inequalities and Applications The remainder of the proof of Theorem 3.7 is similar to that of Theorem 2.3 We, therefore, choose to omit the analogous details involved We thus find that D z ≺ψ z , 3.27 σ which implies that Kξ f ∈ MCλ η, δ; φ, ψ The proof of Theorem 3.7 is thus completed α,β σ Theorem 3.8 Let f ∈ MQCλ η, δ; φ, ψ with φ ∈ P and 3.16 holds Then the function Kξ f ∈ α,β Σ defined by 3.17 belongs to the class MQCλ η, δ; φ, ψ α,β Proof By virtue of 1.26 and Theorem 3.7, and by similarly applying the method of proof of Theorem 3.2, we deduce that the assertion of Theorem 3.8 holds Subordination and Superordination Results In this section, we derive some subordination and superordination results associated with the operator Qλ By similarly applying the methods of proof of the results obtained by Cho α,β et al 15 , we get the following subordination and superordination results Here, we choose to omit the details involved For some other recent sandwich-type results in analytic function theory, one can find in 16–30 and the references cited therein Corollary 4.1 Let f, g ∈ Σ If R zϕ z ϕ z >− z ∈ U; ϕ z : zQλ g z , α,β 4.1 where β : α − 1− β β α , α 4.2 then the subordination relationship zQλ f z ≺ zQλ g z α,β α,β 4.3 implies that zQλ α Furthermore, the function zQλ α 1,β 1,β f z ≺ zQλ α 1,β g g is the best dominant z 4.4 Journal of Inequalities and Applications 15 Corollary 4.2 Let f, g ∈ Σ If R zχ z χ z >− z ∈ U; χ z : zQλ g z , α,β 4.5 where λ2 − − λ2 , 4λ 4.6 zQλ f z ≺ zQλ g z α,β α,β 4.7 zQλ f z ≺ zQλ g z α,β α,β 4.8 : then the subordination relationship implies that Furthermore, the function zQλ g is the best dominant α,β Denote by Q the set of all functions f that are analytic and injective on U − E f , where ε ∈ ∂U : lim f z E f z→ε ∞ , 4.9 and such that f ε / for ε ∈ ∂U − E f If f is subordinate to F, then F is superordinate to f We now derive the following superordination results Corollary 4.3 Let f, g ∈ Σ If R zϕ z ϕ z >− z ∈ U; ϕ z : zQλ g z , α,β where is given by 4.2 , also let the function zQλ f be univalent in U and zQλ α,β α subordination relationship zQλ g z ≺ zQλ f z α,β α,β 4.10 1,β f ∈ Q, then the 4.11 implies that zQλ α Furthermore, the function zQλ α 1,β 1,β g z ≺ zQλ α 1,β f g is the best subordinant z 4.12 16 Journal of Inequalities and Applications Corollary 4.4 Let f, g ∈ Σ If zχ z χ z R >− z ∈ U; χ z : zQλ g z , α,β 4.13 where is given by 4.6 , also let the function zQλ f be univalent in U and zQλ f ∈ Q, then the α,β α,β subordination relationship zQλ g z ≺ zQλ f z α,β α,β 4.14 zQλ g z ≺ zQλ f z α,β α,β 4.15 implies that Furthermore, the function zQλ g is the best subordinant α,β Combining the above mentioned subordination and superordination results involving the operator Qλ , we get the following “sandwich-type results” α,β Corollary 4.5 Let f, gk ∈ Σ R k zϕk z 1, If >− ϕk z z ∈ U; ϕk z : zQλ gk z α,β k 1, , where is given by 4.2 , also let the function zQλ f be univalent in U and zQλ α,β α subordination chain 1,β 4.16 f ∈ Q, then the zQλ g1 z ≺ zQλ f z ≺ zQλ g2 z α,β α,β α,β 4.17 implies that zQλ α 1,β g1 z ≺ zQλ α 1,β f z ≺ zQλ α Furthermore, the functions zQλ α best dominant g 1,β and zQλ α g 1,β are, respectively, the best subordinant and the Corollary 4.6 Let f, gk ∈ Σ R zχk z χk z k 1,β g2 z 4.18 1, If >− z ∈ U; χk z : zQλ gk z α,β k 1, , 4.19 Journal of Inequalities and Applications 17 where is given by 4.6 , also let the function zQλ f be univalent in U and zQλ f ∈ Q, then the α,β α,β subordination chain zQλ g1 z ≺ zQλ f z ≺ zQλ g2 z α,β α,β α,β 4.20 zQλ g1 z ≺ zQλ f z ≺ zQλ g2 z α,β α,β α,β 4.21 implies that Furthermore, the functions zQλ g1 and zQλ g2 are, respectively, the best subordinant and the best α,β α,β dominant Acknowledgments The present investigation was supported by the Scientific Research Fund of Hunan Provincial Education Department under Grant 08C118 of China The authors would like to thank Professor R M Ali for sending several valuable papers to them References I B Jung, Y C Kim, and H M Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications, vol 176, no 1, pp 138–147, 1993 A Y Lashin, “On certain subclasses of meromorphic functions associated with certain integral operators,” 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Analysis and Applications, vol 337, no 1, pp 505–515, 2008 18 Journal of Inequalities and Applications 13 P Eenigenburg, S S Miller, P T Mocanu, and M O Reade, “On a Briot-Bouquet differential subordination,” in General Mathematics 3, vol 64 of International Series of Numerical Mathematics, pp 339348, Birkhă user, Basel, Switzerland, 1983, Revue Roumaine de Math´ matiques Pures et a e Appliqu´ es, vol 29, no 7, pp 567–573, 1984 e 14 S S Miller and P T Mocanu, “Differential subordinations and univalent functions,” The Michigan Mathematical Journal, vol 28, no 2, pp 157–172, 1981 15 N E Cho, O S Kwon, S Owa, and H M Srivastava, “A class of integral operators preserving subordination and superordination for meromorphic functions,” Applied Mathematics and Computation, vol 193, no 2, pp 463–474, 2007 16 R M Ali, V Ravichandran, M H Khan, and K G Subramanian, “Differential sandwich theorems for certain analytic functions,” Far East Journal of Mathematical Sciences, vol 15, no 1, pp 87–94, 2004 17 R M Ali, V Ravichandran, and N Seenivasagan, “Subordination and superordination on Schwarzian derivatives,” Journal of Inequalities and Applications, vol 2008, Article ID 712328, 18 pages, 2008 18 R M Ali, V Ravichandran, and N Seenivasagan, “Subordination and superordination of the LiuSrivastava linear operator on meromorphic functions,” Bulletin of the Malaysian Mathematical Sciences Society, vol 31, no 2, pp 193–207, 2008 19 R M Ali, V Ravichandran, and N Seenivasagan, “Differential subordination and superordination of analytic functions defined by the multiplier transformation,” Mathematical Inequalities & Applications, vol 12, no 1, pp 123–139, 2009 20 T Bulboac˘ , “Sandwich-type theorems for a class of integral operators,” Bulletin of the Belgian a Mathematical Society Simon Stevin, vol 13, no 3, pp 537–550, 2006 21 N E Cho, J Nishiwaki, S Owa, and H M Srivastava, “Subordination and superordination for multivalent functions associated with a class of fractional differintegral operators,” Integral Transforms and Special Functions In press 22 N E Cho and H M Srivastava, “A class of nonlinear integral operators preserving subordination and superordination,” Integral Transforms and Special Functions, vol 18, no 1-2, pp 95–107, 2007 23 S P Goyal, P Goswami, and H Silverman, “Subordination and superordination results for a class of analytic multivalent functions,” International Journal of Mathematics and Mathematical Sciences, vol 2008, Article ID 561638, 12 pages, 2008 24 T N Shanmugam, V Ravichandran, and S Sivasubramanian, “Differential sandwich theorems for some subclasses of analytic functions,” The Australian Journal of Mathematical Analysis and Applications, vol 3, no 1, article 8, 11 pages, 2006 25 T N Shanmugam, S Sivasubramanian, B A Frasin, and S Kavitha, “On sandwich theorems for certain subclasses of analytic functions involving Carlson-Shaffer operator,” Journal of the Korean Mathematical Society, vol 45, no 3, pp 611–620, 2008 26 T N Shanmugam, S Sivasubramanian, and S Owa, “On sandwich results for some subclasses of analytic functions involving certain linear operator,” Integral Transforms and Special Functions In press 27 T N Shanmugam, S Sivasubramanian, and H Silverman, “On sandwich theorems for some classes of analytic functions,” International Journal of Mathematics and Mathematical Sciences, vol 2006, Article ID 29684, 13 pages, 2006 28 T N Shanmugam, S Sivasubramanian, and H M Srivastava, “Differential sandwich theorems for certain subclasses of analytic functions involving multiplier transformations,” Integral Transforms and Special Functions, vol 17, no 12, pp 889–899, 2006 29 Z.-G Wang, R Aghalary, M Darus, and R W Ibrahim, “Some properties of certain multivalent analytic functions involving the Cho-Kwon-Srivastava operator,” Mathematical and Computer Modelling, vol 49, no 9-10, pp 1969–1984, 2009 30 Z.-G Wang, R.-G Xiang, and M Darus, “A family of integral operators preserving subordination and 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Y C Kim, and H M Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications, vol... relationships and integral- preserving properties of certain subclasses of meromorphic functions associated with a family of integral operators,” Journal of Mathematical Analysis and Applications, vol... functions associated with a family of multiplier transformations,” Journal of Mathematical Analysis and Applications, vol 300, no 2, pp 505–520, 2004 R M El-Ashwah and M K Aouf, “Hadamard product of