Báo cáo hóa học: Research Article Subordination and Superordination for Multivalent Functions Associated with the Dziok-Srivastava Operator

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Báo cáo hóa học: " Research Article Subordination and Superordination for Multivalent Functions Associated with the Dziok-Srivastava Operator"Hindawi Publishing CorporationJournal of Inequalities and ApplicationsVolume 2011, Article ID 486595, 17 pagesdoi:10.1155/2011/486595Research ArticleSubordination and Superordination forMultivalent Functions Associated withthe Dziok-Srivastava Operator Nak Eun Cho,1 Oh Sang Kwon,2 Rosihan M. Ali,3 and V. Ravichandran3, 4 1 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea 2 Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea 3 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia 4 Department of Mathematics, University of Delhi, Delhi 110007, India Correspondence should be addressed to Rosihan M. Ali, rosihan@cs.usm.my Received 21 September 2010; Revised 18 January 2011; Accepted 26 January 2011 Academic Editor: P. J. Y. Wong Copyright q 2011 Nak Eun Cho 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. Subordination and superordination preserving properties for multivalent functions in the open unit disk associated with the Dziok-Srivastava operator are derived. Sandwich-type theorems for these multivalent functions are also obtained.1. IntroductionLet Í : {z ∈ : |z| < 1} be the open unit disk in the complex plane , and let H : H Ídenote the class of analytic functions defined in Í. For n ∈ Æ : {1, 2, . . .} and a ∈ , letH a, n consist of functions f ∈ H of the form f z a an zn an 1 zn 1 · · · . Let f and Fbe members of H. The function f is said to be subordinate to F , or F is said to be superordinateto f , if there exists a function w analytic in Í, with |w z | ≤ |z| and such that f z Fwz .In such a case, we write f ≺ F or f z ≺ F z . If the function F is univalent in Í, then f ≺ F if F 0 and f Í ⊂ F Í cf. 1, 2 . Let ϕ : 2 → , and let h be univalent inand only if f 0Í. The subordination ϕ p z , zp z ≺ h z is called a first-order differential subordination.It is of interest to determine conditions under which p ≺ q arises for a prescribed univalentfunction q. The theory of differential subordination in is a generalization of a differentialinequality in Ê, and this theory of differential subordination was initiated by the works ofMiller, Mocanu, and Reade in 1981. Recently, Miller and Mocanu 3 investigated the dual2 Journal of Inequalities and Applicationsproblem of differential superordination. The monograph by Miller and Mocanu 1 gives agood introduction to the theory of differential subordination, while the book by Bulboac˘ 4 ainvestigates both subordination and superordination. Related results on superordination canbe found in 5–23 . By using the theory of differential subordination, various subordination preservingproperties for certain integral operators were obtained by Bulboac˘ 24 , Miller et al. 25 , aand Owa and Srivastava 26 . The corresponding superordination properties and sandwich-type results were also investigated, for example, in 4 . In the present paper, we investigatesubordination and superordination preserving properties of functions defined through theuse of the Dziok-Srivastava linear operator Hp,q,s α1 see 1.9 and 1.10 , and also obtaincorresponding sandwich-type theorems. The Dziok-Srivastava linear operator is a particular instance of a linear operatordefined by convolution. For p ∈ Æ , let Ap denote the class of functions ∞ zp ak p zk p fz 1.1 k1that are analytic and p-valent in the open unit disk Í with f p1 0 / 0. The Hadamardproduct or convolution f ∗ g of two analytic functions ∞ ∞ ak zk , ...