%0 Journal Article
%T Subclass of Multivalent Harmonic Functions Defi£¿ned by Wright Generalized Hypergeometric Functions
%A M. K. Aouf
%A A. O. Moustafa
%A E. A. Adwan
%J Journal of Complex Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/397428
%X We introduce a new class of multivalent harmonic functions defi£¿ned by Wright generalized hypergeometric function. Coefficient estimates, extreme points, distortion bounds, and convex combination for functions belonging to this class are obtained. 1. Introduction A continuous complex-valued function defined in a simply connected complex domain is said to be harmonic in if both and are real harmonic in . In any simply connected domain the function can be written in the form where and are analytic in , is called the analytic part, and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and sense-preserving in is that in (see [1]). Denote by the class of functions of the form (1) that are harmonic univalent and sense preserving in the unit disc for which . Recently, Ahuja and Jahangiri [2] defined the class consisting of all -valent harmonic functions that are sense-preserving in and , are of the form Let and be positive real parameters such that The Wright generalized hypergeometric function [3] (see also [4]) is defined by If and , we have the relationship where is the generalized hypergeometric function (see [4]) and By using the generalized hypergeometric function, Dziok and Srivastava [5] introduced a linear operator. Dziok and Raina in [6] and Aouf and Dziok in [7] extended this linear operator by using Wright generalized hypergeometric function. Aouf et al. [8] defined the linear operator by the Hadamard product as where is given by We observe that, for a function of the form (2), we have where is given by (7) and is defined by For convenience, we write Now we can define the modified Wright operator as follows: where For , , where is the modified Wright generalized hypergeometric functions (see [9]). We note that, for and , we obtain , where is the modified Dziok-Srivastava operator (see [10]). For , and for all , let denote the family of harmonic -valent functions , where and are given by (2) and satisfying the analytic criterion Let be the subclass of consisting of functions such that and are of the form We note that for suitable choices of and , we obtain the following subclasses:(1) (see [2]);(2) (see [10]);(3) (see [9]);(4) (see [11, 12]);(5) (see [13]). 2. Coefficient Estimates Unless otherwise mentioned, we will assume in the reminder of this paper that the parameters and are positive real numbers, , , , is defined by (7), and is defined by (11). Theorem 1. Let be such that and are given by (2). Furthermore, let Then is orientation preserving in and . Proof. The inequality is enough to show that £¿£¿is
%U http://www.hindawi.com/journals/jca/2013/397428/