The Scientific World Journal

Volume 2013 (2013), Article ID 164287, 7 pages

http://dx.doi.org/10.1155/2013/164287

Research Article

On Harmonic Meromorphic Functions Associated with Basic Hypergeometric Functions

Huda Al dweby and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor D. Ehsan, Malaysia

Received 4 August 2013; Accepted 14 September 2013

Academic Editors: A. Favini and N. Kallur

Copyright © 2013 Huda Al dweby and Maslina Darus. 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.

By making use of basic hypergeometric functions, a class of complex harmonic meromorphic functions with positive coefficients is introduced. We obtain some properties such as coefficient inequality, growth theorems, and extreme points.

1. Introduction and Preliminaries

Recently, the popularity of the study of basic hypergeometric series (also called -hypergeometric series) is increasing among the researchers. Back in 1748, Euler considered the infinite product and ever since it became very important to many areas. However, it was stated in the literature that the development of these functions was slower until Heine (1878) converted a simple observation such that , which then returns the theory of  basic hypergeometric series to the famous theory of Gauss  hypergeometric series. Various authors [1–7] introduced classes of analytic functions involving hypergeometric functions and investigated their properties. However, all these results concern mostly the Gaussian and generalized hypergeometric functions. It seems that no attempt has been made to derive similar results for the basic hypergeometric functions. In this work, we proceed to define a class of harmonic meromorphic functions in the unit disk associated with the basic hypergeometric function and discuss some of its properties.

The subject of harmonic univalent functions is a recent area of research which was initially established by Clunie and Sheil-Small [8]. The importance of these functions is due to their use in the study of minimal surfaces as well as in various problems related to applied mathematics and perhaps to other areas of sciences. Hengartner and Schober [9] have introduced and studied special classes of harmonic functions, which are defined on the exterior of the unit disk . They have proved that these functions are complex valued harmonic, sense preserving, univalent mappings , admitting the representation

where and are analytic in .

For , let denote the class of functions

which are harmonic in the punctured unit disk , where and are analytic in and , respectively. The class was studied in [10–12].

We further denote by the subclass of consisting of functions of the form

which are univalent harmonic in the punctured unit disk .

A function is said to be in the class of meromorphically harmonic starlike functions in if it satisfies the condition

this class has been studied by Jahangiri and Silverman (see [10]). For harmonic functions

the convolution is defined by

For complex parameters , , , the basic hypergeometric function is defined as follows:

with , where when , , denotes the set of positive integers, and is the -shifted factorial defined by

By using the ratio test, one recognizes that if , the series (7) converges absolutely (see [13]) for all if and for if .

The -derivative of a function is defined by

For a function , observe that

then , where is the ordinary derivative. For more properties of , see [14, 15].

Now for , , and , the basic hypergeometric function defined in (7) takes the form

which converges absolutely in the open unit disk .

Let

where, for convenience,

Corresponding to the function , defined in (12), consider

and its inverse function defined by

The series expansion of the inverse function is given as

where

Now, let

which is analytic function in the punctured unit disk .

Again, corresponding to the function , defined in (12), consider

and its inverse function defined by

The series expansion of the inverse function is given as

Let

which is analytic function in . Now, we define the harmonic meromorphic function as

where and are of the forms (18) and (22).

Using convolution of harmonic meromorphic functions given by (23) and given by (2), a linear operator is defined as

Involving the operator , we introduce the class as follows.

Definition 1. Let denote the family of harmonic meromorphic functions satisfying

Equivalently,

for some and for all .

For , , , and , is the class of harmonic meromorphic starlike functions of order and for , , , , and , the class gives the class of harmonic meromorphic starlike functions for all .

Denote .

2. Main Results

Theorem 2. If is of the form (2) and satisfies the condition

where , then is harmonic univalent sense preserving in and .

Proof. Suppose that (27) holds true for . Consider the expression

That is,

The inequality in (28) holds true for all . Therefore, letting in (28), we obtain

By hypothesis (27), it follows that (26) holds, so that . Note that is sense preserving in . This is because

Then, we have . Hence the theorem.

Now, we prove that condition (27) is necessary for functions in .

Theorem 3. Let be a function defined by (3). Then, if and only if the inequality

is satisfied.

Proof. In view of Theorem 2, it suffices to show that the “only if" part is true. Assume that . Then,

Since for all , it follows from (33) that

We now choose the values of on the real axis. Upon the clearing the denominator in (34) and letting through real values, we obtain the following:

which immediately yields the required condition (32).

Next, we consider a distortion property for functions in the class as follows.

Theorem 4. If defined by (3) is in the class , then for , where , , and , one has

Proof. We will only prove the right-hand inequality. The proof for the left-hand inequality is similar and we will omit it:

The functions

for show that the bounds given in Theorem 4 are sharp in .

Next, we give the following.

Theorem 5. Set

Then, , if and only if, it can be expressed in the form

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

Proof. Let

Then,

So, . Conversely, suppose that . Set

Then, by Theorem 3, and .

and note that, by Theorem 3, . Consequently, we obtain

This proves the theorem.

Remark 6. Other work related to -hypergeometric functions and analytic functions can be found in [16, 17].

Acknowledgment

The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02 and UKM-DLP-2011-050.

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