Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 147069, 12 pages doi:10.1155/2009/147069 Research Article Inclusion Properties for Certain Classes of Meromorphic Functions Associated with a Family of Linear Operators Nak Eun Cho Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea Correspondence should be addressed to Nak Eun Cho, necho@pknu.ac.kr Received 3 March 2009; Accepted 1 May 2009 Recommended by Ramm Mohapatra The purpose of the present paper is to investigate some inclusion properties of certain classes of meromorphic 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. Copyright q 2009 Nak Eun Cho. 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. 1. Introduction Let A be the class of analytic functions in the open unit disk U {z ∈ C : |z| < 1} with the usual normalization f0 f 0 − 1 0. If f and g are analytic in U, we say that f is subordinate to g, written f ≺ g or fz ≺ gz, if there exists an analytic function w in U with w0 0 and |wz| < 1 for z ∈ U such that fz gwz. Let N be the class of all functions φ which are analytic and univalent in U and for which φU is convex with φ0 1 and Re{φz} > 0 for z ∈ U. We denote by S ∗ and K the subclasses of A consisting of all analytic functions which are starlike and convex, respectively. Let M denote the class of functions of the form fz ∞ 1 ak zk , z k0 1.1 which are analytic in the punctured open unit disk D U\{0}. For 0 ≤ η, β < 1, we denote by MSη, MKη and MCη, β the subclasses of M consisting of all meromorphic functions which are, respectively, starlike of order η, convex of order η and colse-to-convex of order β and type η in U see, for details, 1, 2. 2 Journal of Inequalities and Applications Making use of the principle of subordination between analytic functions, we introduce the subclasses MSη, φ, MKη, φ and MCη, β; φ, ψ of the class M for 0 ≤ η, β < 1 and φ, ψ ∈ N, which are defined by zf z 1 − − η ≺ φz in U , 1−η fz zf z 1 MK η; φ : f ∈ M : − 1 − η ≺ φz in U , 1−η f z MS η; φ : MC η, β; φ, ψ : f ∈M: zf z 1 − − β ≺ ψz in U . f ∈ M : ∃g ∈ MS η; φ s.t. 1−β gz 1.2 We note that the classes mentioned above are the familiar classes which have been used widely on the space of analytic and univalent functions in U see 3–5 and for special choices for the functions φ and ψ involved in these definitions, we can obtain the well-known subclasses of M. For examples, we have 1z MS η , MS η; 1−z 1z MK η; MK η , 1−z 1z 1z , MC η, β; MC η, β . 1−z 1−z 1.3 Now we define the function φa, c; z by φa, c; z : ∞ ak1 k 1 z , z k0 ck1 z ∈ U; a ∈ R; c ∈ R \ Z−0 ; Z−0 : {−1, −2, . . .} , 1.4 1.5 where νk is the Pochhammer symbol or the shifted factorial defined in terms of the Gamma function by ⎧ Γν k ⎨1, νk : ⎩νν 1 · · · ν k − 1, Γν if k 0, ν ∈ C \ {0}, if k ∈ N : {1, 2, . . .}, ν ∈ C. 1.6 Let f ∈ M. Denote by La, c : M → M the operator defined by La, cfz φa, c; z ∗ fz z ∈ D, 1.7 where the symbol ∗ stands for the Hadamard product or convolution. The operator La, c was introduced and studied by Liu and Srivastava 6. Further, we remark in passing that this Journal of Inequalities and Applications 3 operator La, c is closely related to the Carlson-Shaffer operator 7 defined on the space of analytic and univalent functions in U. Corresponding to the function φa, c; z, let φ† a, c; z be defined such that φa, c; z ∗ φλ a, c; z 1 z1 − zλ λ > 0. 1.8 Analogous to La, c, we now introduce a linear operator Lλ a, c on M as follows: Lλ a, cfzc φλ a, c; z ∗ fz, 1.9 a, c ∈ R \ Z−0 ; λ > 0; z ∈ U; f ∈ M . 1.10 We note that L1 2, 1fz fz, L1 1, 1fz zf z 2fz. 1.11 We note that the operator Lλ a, c is motivated essentially to the integral operator for analytic functions defined by Choi et al. 3, which extends the Noor integral operator studied by K. I. Noor and M. A. Noor 8 also, see 9–13. Next, by using the operator Lλ a, c, we introduce the following classes of meromprphic functions for φ, ψ ∈ N, a, c ∈ R \ Z−0 , λ > 0 and 0 ≤ η, β < 1: MSλa,c η; φ : f ∈ M : Lλ a, cf ∈ MS η; φ , λ MKa,c η; φ : f ∈ M : Lλ a, cf ∈ MK η; φ , 1.12 λ MCa,c η, β; φ, ψ : f ∈ M : Lλ a, cf ∈ MC η, β; φ, ψ . We also note that λ fz ∈ MKa,c η; φ ⇐⇒ −zf z ∈ MSλa,c η; φ . 1.13 In particular, we set 1 Az MSλa,c η; A, B η; −1 < B < A ≤ 1, 1 Bz 1 Az λ λ MKa,c η; MKa,c η; A, B −1 < B < A ≤ 1. 1 Bz MSλa,c 1.14 In this paper, we investigate several inclusion properties of the classes MSλa,c η; φ, λ and MCa,c η; φ associated with the operator Lλ a, c defined by 1.9.Some λ η; φ, MKa,c 4 Journal of Inequalities and Applications invariant properties under convolution are also considered for the classes mentioned above. Furthermore, relevant connections of the results presented here with those obtained in earlier works are pointed out. 2. Inclusion Properties Involving the Operator Lλ a, c The following lemmas will be required in our investigation. Lemma 2.1. Let φλi a, c; z, φλ ai , c; z, and φλ a, ci ; z i 1, 2 be defined by 1.9. Then for λi > 0, ai , ci ∈ R \ Z−0 i 1, 2, φλ1 a, c; z φλ2 a, c; z∗fλ1 ,λ2 z, 2.1 φλ a2 , c; z φλ a1 , c; z∗fa1 ,a2 z, 2.2 φλ a, c1 ; z φλ a, c2 ; z∗fc1 ,c2 z, 2.3 where fs,t z ∞ sk1 k 1 z z k0 tk1 z ∈ D. 2.4 Proof. From 1.8, we know that ∞ ck1 λk1 k 1 z z k0 ak1 1k1 φλ a, c; z z ∈ D. 2.5 Therefore 2.1, 2.2 and 2.3 follow from 2.5 immediately. Lemma 2.2 see 14, pages 60–61. Let t ≥ s > 0. If t ≥ 2 or s t ≥ 3, then the function z2 fs,t z belongs to the class K, where fs,t is defined by 2.4. Lemma 2.3 see15. Let f ∈ K and g ∈ S∗ . Then for every analytic function h in U, f∗hg U ⊂ cohU, f∗g U 2.6 where cohU denote the closed convex hull of hU. λ η; φ is contained in Theorem 2.4. At first, the inclusion relationship involving the class MSa,c Theorem 2.4. Let λ2 ≥ λ1 > 0, a, c ∈ R \ Z−0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If λ2 ≥ 2 or λ1 λ2 ≥ 3, then λ2 λ1 MSa,c η; φ ⊂ MSa,c η; φ . 2.7 Journal of Inequalities and Applications 5 λ2 λ2 η; φ. From the definition of MSa,c η; φ, we have Proof. Let f ∈ MSa,c 1 1−η z Lλ2 a, cfz − η φwz − Lλ2 a, cfz z ∈ U, z z2 Lλ2 a, cfz 1z 2 − 1 − η φwz − η ≺ 2 1−z z Lλ2 a, cfz z ∈ U, 2.8 2.9 where w is analytic in U with |wz| < 1 z ∈ U and w0 0 φ0 − 1. By using 1.9, 2.1 and 2.8, we get z φλ1 a, c; z ∗ fz z Lλ1 a, cfz − − Lλ1 a, cfz φλ1 a, c; z ∗ fz z φλ2 a, c; z ∗ fλ1 ,λ2 z ∗ fz − φλ2 a, c; z ∗ fλ1 ,λ2 z ∗ fz fλ1 ,λ2 z ∗ −z Lλ2 a, cfz fλ1 ,λ2 z ∗ Lλ2 a, cfz fλ1 ,λ2 z ∗ 1 − η φwz η Lλ2 a, cfz . fλ1 ,λ2 z ∗ Lλ2 a, cfz 2.10 Therefore by using 2.8, we obtain zLλ1 a, cfz − −η Lλ1 a, cfz fλ1 ,λ2 z ∗ 1 − η φwz η Lλ2 a, cfz 1 −η 1−η fλ1 ,λ2 z ∗ Lλ2 a, cfz 2 z fλ1 ,λ2 z ∗ 1 − η φwz η z2 Lλ2 a, cfz 1 −η . 1−η z2 fλ1 ,λ2 z ∗ z2 Lλ2 a, cfz 1 1−η 2.11 It follows from 2.9 and Lemma 2.2 that z2 Lλ2 a, cfz ∈ S∗ and z2 fλ1 ,λ2 ∈ K, respectively. Let us put swz : 1 − ηφwz η. Then by applying Lemma 2.3 to 2.10, we obtain 2 z fλ1 ,λ2 ∗ swzz2 Lλ2 a, cf 2 U ⊂ coswU ⊂ sU, z fλ1 ,λ2 ∗ z2 Lλ2 a, cf 2.12 6 Journal of Inequalities and Applications since s is convex univalent. Therefore from the definition of subordination and 2.12, we have 1 1−η z Lλ1 a, cfz − − η ≺ φz Lλ1 a, cfz z ∈ U, 2.13 λ1 or, equivalently, f ∈ MSa,c φ, which completes the proof of Theorem 2.4. By using 1.13, 2.2 and 2.3, we have the following Theorem 2.5 and Theorem 2.6. Theorem 2.5. Let λ > 0, a2 ≥ a1 > 0, c ∈ R \ Z−0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If a2 ≥ 2 or a1 a2 ≥ 3, then MSaλ1 ,c η; φ ⊂ MSaλ2 ,c η; φ . 2.14 Theorem 2.6. Let λ > 0, a ∈ R \ Z−0 , c2 ≥ c1 > 0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If c2 ≥ 2 or c1 c2 ≥ 3, then λ λ MSa,c η; φ ⊂ MSa,c η; φ . 2 1 2.15 Next, we prove the inclusion theorem involving the class MKλa,c η; φ. Theorem 2.7. Let λ2 ≥ λ1 > 0, a, c ∈ R \ Z−0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If λ2 ≥ 2 or λ1 λ2 ≥ 3, then 2 1 MKλa,c η; φ ⊂ MKλa,c η; φ . 2.16 Proof. Applying 1.13 and Theorem 2.4, we observe that 2 fz ∈ MKλa,c η; φ ⇐⇒ Lλ2 a, cfz ∈ MK η; φ ⇐⇒ −z Lλ2 a, cfz ∈ MS η; φ ⇐⇒ Lλ2 a, c −zf z ∈ MS η; φ λ2 ⇐⇒ −zf z ∈ MSa,c η; φ λ1 ⇒ −zf z ∈ MSa,c η; φ ⇐⇒ Lλ1 a, c −zf z ∈ MS η; φ ⇐⇒ −z Lλ1 a, cfz ∈ MS η; φ ⇐⇒ Lλ1 a, cfz ∈ MK η; φ 1 ⇐⇒ fz ∈ MKλa,c η; φ , which evidently proves Theorem 2.7. 2.17 Journal of Inequalities and Applications 7 By using a similar method as in the proof of Theorem 2.7, we obtain the following two theorems. Theorem 2.8. Let λ > 0, a2 ≥ a1 > 0, c ∈ R \ Z−0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If a2 ≥ 2 or a1 a2 ≥ 3, then MKλa1 ,c η; φ ⊂ MKλa2 ,c η; φ . 2.18 Theorem 2.9. Let λ > 0, a ∈ R \ Z−0 , c2 ≥ c1 > 0 and φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If c2 ≥ 2 or c1 c2 ≥ 3, then MKλa,c2 η; φ ⊂ MKλa,c1 η; φ . 2.19 Taking φz 1 Az/1 Bz −1 < B < A ≤ 1; z ∈ U in Theorems 2.4–2.9, we have the following corollaries below. Corollary 2.10. Let 1 A1 − η < 2 − η1 B −1 < B < A ≤ 1; 0 ≤ η < 1 and c ∈ R \ Z−0 . If λ2 ≥ λ1 > 0 and λ2 ≥ min{2, 3 − λ1 }, and a2 ≥ a1 > 0 and a2 ≥ min{2, 3 − a1 }, then MSaλ12,c η; A, B ⊂ MSaλ11,c η; A, B ⊂ MSaλ21,c η; A, B , MKλa21 ,c η; A, B ⊂ MKλa11 ,c η; A, B ⊂ MKλa12 ,c η; A, B . 2.20 Corollary 2.11. Let 1 A1 − η < 2 − η1 B −1 < B < A ≤ 1; 0 ≤ η < 1 and λ > 0. If a2 ≥ a1 > 0 and a2 ≥ min{2, 3 − a1 }, and c2 ≥ c1 > 0 and c2 ≥ min{2, 3 − c1 }, then MSaλ1 ,c2 η; A, B ⊂ MSaλ1 ,c1 η; A, B ⊂ MSaλ2 ,c1 η; A, B , MKλa1 ,c2 η; A, B ⊂ MKλa1 ,c1 η; A, B ⊂ MKλa2 ,c1 η; A, B . 2.21 Corollary 2.12. Let 1 A1 − η < 2 − η1 B −1 < B < A ≤ 1; 0 ≤ η < 1 and a ∈ R \ Z−0 . If λ2 ≥ λ1 > 0 and λ2 ≥ min{2, 3 − λ1 }, and c2 ≥ c1 > 0 and c2 ≥ min{2, 3 − c1 }, then λ2 λ2 λ1 MSa,c η; A, B ⊂ MSa,c η; A, B ⊂ MSa,c η; A, B , 2 1 1 2 2 1 MKλa,c η; A, B ⊂ MKλa,c η; A, B ⊂ MKλa,c η; A, B . 2 1 1 2.22 To prove theorems below, we need the following lemma. Lemma 2.13. Let φ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If f ∈ M with z2 fz ∈ K and q ∈ MSη; φ, then f ∗ q ∈ MSη; φ. 8 Journal of Inequalities and Applications Proof. Let q ∈ MSη; φ. Then −zq z 1 − η φwz η qz z ∈ U, 2.23 where w is an analytic function in U with |wz| < 1 z ∈ U and w0 0. Thus we have 1 1−η z fz ∗ qz − −η fz ∗ qz 1 1−η 1 1−η fz ∗ −zq z −η fz ∗ qz fz ∗ 2.24 1 − η φωz η qz −η z ∈ D. fz ∗ qz By using the similar arguments to those used in the proof of Theorem 2.4, we conclude that 2.24 is subordinated to φ in U and so f ∗ q ∈ MSη; φ. Finally, we give the inclusion properties involving the class MCλa,c η, β; φ, ψ. Theorem 2.14. Let c ∈ R \ Z−0 and φ, ψ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If λ2 ≥ λ1 > 0 and λ2 ≥ min{2, 3 − λ1 }, and a2 ≥ a1 > 0 and a2 ≥ min{2, 3 − a1 }, then MCλa21 ,c η, β; φ, ψ ⊂ MCλa11 ,c η, β; φ, ψ ⊂ MCλa12 ,c η, β; φ, ψ . 2.25 Proof. We begin by proving that MCλa21 ,c η, β; φ, ψ ⊂ MCλa11 ,c η, β; φ, ψ . 2.26 Let f ∈ MCλa21 ,c η, β; φ, ψ. Then there exists a function q2 ∈ MSη; φ such that 1 1−β z Lλ2 a1 , cfz − − β ≺ ψz q2 z 0 ≤ β < 1; z ∈ U . 2.27 From 2.27, we obtain −z Lλ2 a1 , cfz 1 − β ψwz β q2 z, 2.28 Journal of Inequalities and Applications 9 where w is an analytic function in U with |wz| < 1 z ∈ U and w0 0. By virtue of 2.3, Lemmas 2.2 and 2.13, we see that fλ1 ,λ2 z ∗ q2 z ≡ q1 z belongs to MSη; φ. Then, making use of 2.1, we have 1 1−β z Lλ1 a1 , cfz − −β q1 z ⎛ ⎞ f , cfz ∗ −z L z a λ ,λ λ 1 1 2 2 1 ⎜ ⎟ − β⎠ ⎝ 1−β fλ1 ,λ2 z ∗ q2 z 1 1−β 1 1−β ≺ ψz fλ1 ,λ2 z ∗ 1 − β ψwz β q2 z −β fλ1 ,λ2 z ∗ q2 z 2.29 z2 fλ1 ,λ2 z ∗ 1 − β ψwz β z2 q2 z −β z2 fλ1 ,λ2 z ∗ z2 q2 z z ∈ U. Therefore we prove that f ∈ MCλa11 ,c η, β; φ, ψ. For the second part, by using arguments similar to those detailed above with 2.2, we obtain MCλa11 ,c η, β; φ, ψ ⊂ MCλa12 ,c η, β; φ, ψ 2.30 Thus the proof of Theorem 2.14 is completed. The following results can be obtained by using the same techniques as in the proof of Theorem 2.14 and so we omit the detailed proofs involved. Theorem 2.15. Let λ > 0 and φ, ψ ∈ N with Re{φz} < 2−η/1−η 0 ≤ η < 1. If a2 ≥ a1 > 0 and a2 ≥ min{2, 3 − a1 }, and c2 ≥ c1 > 0 and c2 ≥ min{2, 3 − c1 }, then MCλa1 ,c2 η, β; φ, ψ ⊂ MCλa1 ,c1 η, β; φ, ψ ⊂ MCλa2 ,c1 η, β; φ, ψ . 2.31 Theorem 2.16. Let a ∈ R \ Z−0 and φ, ψ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1. If λ2 ≥ λ1 > 0 and λ2 ≥ min{2, 3 − λ1 }, and c2 ≥ c1 > 0 and c2 ≥ min{2, 3 − c1 }, then 2 2 1 MCλa,c η, β; φ, ψ ⊂ MCλa,c η, β; φ, ψ ⊂ MCλa,c η, β; φ, ψ . 2 1 1 2.32 Remark 2.17. For a λ 1 λ > −1 and c 1, Theorems 2.4, 2.5, 2.7, 2.8, and 2.14 reduce to the corresponding results obtained by Cho and Noor 16. 3. Inclusion Properties Involving Various Operators λ The next theorem shows that the classes MSa,c η; φ, MKλa,c η; φ and MCλa,c η, β; φ, ψ are invariant under convolution with convex functions. 10 Journal of Inequalities and Applications Theorem 3.1. Let λ > 0, a > 0, c ∈ R \ Z−0 , φ, ψ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1 and let g ∈ M with z2 gz ∈ K. Then λ λ η; φ ⇒ g∗f ∈ MSa,c η; φ, i f ∈ MSa,c ii f ∈ MKλa,c η; φ ⇒ g∗f ∈ MKλa,c η; φ, iii f ∈ MCλa,c η, β; φ, ψ ⇒ g∗f ∈ MCλa,c η, β; φ, ψ. λ Proof. i Let f ∈ MSa,c η; φ. Then we have 1 1−η ⎛ ⎞ gz ∗ −z L cfz a, λ z Lλ a, c g ∗ f z 1 ⎜ ⎟ − η⎠. −η − ⎝ 1−η gz ∗ Lλ a, cfz Lλ a, c g ∗ f z 3.1 By using the same techniques as in the proof of Theorem 2.4, we obtain i. ii Let f ∈ MKλa,c φ. Then, by 1.13, −zf z ∈ MSa,c η; φ and hence from i, λ η; φ. Since gz ∗ −zf z ∈ MSa,c gz ∗ −zf z −z g ∗ f z, 3.2 we have ii applying 1.13 once again. iii Let f ∈ MCλa,c η, β; φ, ψ. Then there exists a function q ∈ MSη; φ such that −z Lλ a, cfz 1 − β ψwz β qz 0 ≤ β < 1; z ∈ U , 3.3 where w is an analytic function in U with |wz| < 1 z ∈ U and w0 0. From Lemma 2.13, we have that g ∗ q ∈ MSη; φ. Since 1 1−β z Lλ a, c g ∗ f z − −β g ∗ q z ⎛ ⎞ gz ∗ −z L cfz a, λ 1 ⎜ ⎟ − β⎠ ⎝ 1−β gz ∗ qz 1 1−β ≺ ψz we obtain iii. z2 gz ∗ z ∈ U, 1 − β ψwz β z2 qz −β z2 gz ∗ z2 qz 3.4 Journal of Inequalities and Applications 11 Now we consider the following operators defined by ∞ 1 c k−1 1 z Re{c} ≥ 0; z ∈ D, z k1 k c 1 − xz 1 log Ψ2 z 2 log 1 0; |x| ≤ 1, x / 1; z ∈ D . 1−z z 1 − x Ψ1 z 3.5 It is well known 17 that the operators z2 Ψ1 and z2 Ψ2 are convex univalent in U. Therefore we have the following result, which can be obtained from Theorem 3.1 immediately. Corollary 3.2. Let a > 0, λ > 0, c ∈ R \ Z−0 , φ, ψ ∈ N with Re{φz} < 2 − η/1 − η 0 ≤ η < 1 and let Ψi i 1, 2 be defined by 3.5, respectively. 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