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Journal of Complex Analysis
Volume 2013, Article ID 397428, 7 pages
http://dx.doi.org/10.1155/2013/397428
Research Article

Subclass of Multivalent Harmonic Functions Defi…ned by Wright Generalized Hypergeometric Functions

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 13 May 2013; Revised 10 September 2013; Accepted 10 September 2013

Academic Editor: J. Dziok

Copyright © 2013 M. K. Aouf 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.

Abstract

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 orientation preserving. Note that Now we will show that . We only need to show that if (17) holds, then condition (15) is satisfied. Using the fact that if and only if , it suffices to show that where Substituting for and in (17) yields The last expression is nonnegative by (17). This completes the proof of Theorem 1. The harmonic -valent function where , shows that the coefficient bound given by (17) is sharp. It is worthy to note that the function of the form (22) belongs to the class for all because coefficient inequality (17) holds.

Theorem 2. A function is in the class , if and only if

Proof. From (15), we have the necessary and sufficient condition for that is, Hence we have the equivalent condition , . Simple algebraic manipulation of (25) yields This completes the proof of Theorem 2.

Theorem 3. A function , where and are given by (16), is in the class , if and only if

Proof . Since , we only need to prove the “only if” part of this theorem. To this end, for functions , where and given by (16), we notice that condition is equivalent to The above condition must hold for all , . Choosing the values of on the positive real axis where , we must have If condition (27) does not hold, then the numerator in (30) is negative for sufficiently close to . Hence there exist in for which the quotient in (30) is negative. This contradicts the required condition for . This completes the proof of Theorem 3.

3. Distortion Theorem

Theorem 4. Let the function , where and are given by (16), belong to the class . Then for , we have for . The results are sharp with equality for the functions defined by

Proof. We only prove the first inequality. The proof for the second inequality is similar and will be omitted. Let . Taking the absolute value of , we have This completes the proof of Theorem 4.

Putting and in Theorem 4, we obtain the following corollary which modifies the result obtained by Omar and Halim [10, Theorem 2.6].

Corollary 5. Let the function , where and are given by (16), belong to the class . Then, for , we have for . The results are sharp with equality for the functions defined by where with and .

4. Extreme Points

Theorem 6. Let , where and are given by (16). Then , if and only if where , , , . In particular, the extreme points of the class are and .

Proof. Suppose that Then and so .
Conversely, if , then Set Since    and , , then we can see that can be expressed in the form (37). This completes the proof of Theorem 6.

Now we show that the class is closed under convex combinations of its members.

Theorem 7. The class is closed under convex combination.

Proof. For , let , where is given by Then by using Theorem 3, we have For , , the convex combination of may be written as Then by (45), we have This is the required condition and so . This completes the proof of Theorem 7.

Acknowledgment

The authors would like to thank the referees of the paper for their helpful suggestions.

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