Journal of Function Spaces
Volume 2016 (2016), Article ID 7123907, 6 pages
http://dx.doi.org/10.1155/2016/7123907

Research Article

A Subclass of Harmonic Functions Related to a Convolution Operator

Saqib Hussain,¹ Akhter Rasheed,¹ and Maslina Darus²

¹Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad Campus, Abbottabad, Pakistan
²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 23 May 2016; Accepted 21 July 2016

Academic Editor: Wilfredo Urbina

Copyright © 2016 Saqib Hussain 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 subclass of harmonic functions by using a certain linear operator. For this class we derive coefficient bounds, extreme points, and inclusion results and also show that this class is closed under an integral operator.

1. Introduction

Harmonic functions have long been used in the representation of minimal surface; for example, Heinz [1] in 1952 used such mappings in the study of the Gaussian curvature of nonparametric minimal surface over the unit disc. Such mappings have vast application in the field of engineering, physics, electronics, medicine, operations research, aerodynamics, and other branches of applied mathematics (see [2]).

is said to be complex valued harmonic function if both and are continuous and real harmonic; that is, and . In simply connected domain [3], we can write , where and are analytic in We call the analytic part and the coanalytic part of . The necessary and sufficient condition for a function to be univalent and sense preserving in is (see [3]). A function is in class if it is harmonic, univalent, and sense preserving in , where and have the following series:

Hence

Note that if the coanalytic part is identically zero, then reduces to well-known class of normalized univalent analytic functions For this class, the function may be expressed as

A function given by (2) is said to be harmonic starlike of order for , if

or equivalently

The class of all harmonic starlike functions of order is denoted by This class was studied by Jahangiri [4]. The cases and were studied by Silverman and Silvia [5] and Silverman [6].

For and given by (2), the convolution is denoted by and defined as

Let and be positive real parameter such that

The Wright generalized hypergeometric function [7] is defined as

which is defined by

If and , we have the relationship

is the generalized hypergeometric function [8], where denotes the set of all positive integers and is the Pochhammer symbol and

Dziok and Srivastava [8] introduced a linear operator which is generalization of Dziok-Srivastava operator [8–10], Carlson-Shaffer operator [11], and the generalized Bernardi’s integral operator [12].

Dziok and Raina [13] considered the linear operator defined by

where is given by

For , we have

where is given by

For our convenience we write

Motivated by the work of [4, 14–21], we extend the work of Chandrashekar et al. [22] by introducing some new subclasses of using the generalized hypergeometric function.

Definition 1. A function is in class if

Definition 2. Let denote the class of functions of the form

and .

Throughout this paper, we shall assume , , , and as given in (11) and (15), respectively, unless otherwise mentioned.

2. Main Results

In Theorem 3, we shall present a sufficient condition for to be in class

Theorem 3. Let be given by (2). If

then

Proof. When inequality (18) holds for the coefficients of given in (2), we have to show that inequality (16) is satisfied. Arranging the left side inequality (16), we have

As we know if and only if , it is sufficient to show that

Substituting the values and in left side of (20), we obtain

by hypothesis in (18), which implies that

Now we obtain the necessary and sufficient condition for the function given by (16) to be in

Theorem 4. Let be given by (16). Then if and only if

Proof. Since , we only have to prove the necessary part of theorem. Assume that , and then by virtue of (16), we obtain

This is equivalent to

This condition must hold for all values of and for real , so that, by taking and , the above inequality reduces to

Letting through real values, we obtain condition (22). This completes the proof.

We determine the extreme points of closed convex hulls of denoted by

Theorem 5. A function if and only if

where

, , and In particular, the extreme points of are and

Proof. First, we consider

where , and
Using (22) for the coefficients in (29), we have

and hence
Conversely, suppose that , and set

where Then

which is the required result.

Next we show that is close under convex combinations of its members.

Theorem 6. The family is closed under convex combination.

Proof. For , suppose that , where

For coefficients in (22) the relation given in (33) takes the form

For , , the convex combination of may be written as

From inequality (22) for (35),

and therefore

Theorem 7. For , let and Then

Proof. Let

and then

We note that and . Therefore

by using (22), since and . This proves that .

Now for the class the closure property under the generalized Bernardi-Livingston integral operator is examined which is defined by

Theorem 8. Let and then

Proof. Consider the generalized Bernardi-Livingston integral operator given in (41):

where

Therefore

Since , therefore, by Theorem 4,

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by AP-2013-009 and GP-K006392, Universiti Kebangsaan Malaysia.

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