International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 374821, 6 pages

http://dx.doi.org/10.1155/2014/374821

Research Article

A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized Hypergeometric Functions

F. Ghanim¹ and M. Darus²

¹Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, UAE

²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 16 May 2014; Accepted 21 June 2014; Published 9 July 2014

Academic Editor: Hari M. Srivastava

Copyright © 2014 F. Ghanim and M. 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.

We introduce a new class of meromorphically analytic functions, which is defined by means of a Hadamard product (or convolution) involving some suitably normalized meromorphically functions related to Cho-Kwon-Srivastava operator. A characterization property giving the coefficient bounds is obtained for this class of functions. The other related properties, which are investigated in this paper, include distortion and the radii of starlikeness and convexity. We also consider several applications of our main results to generalized hypergeometric functions.

1. Introduction

A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic function is a function of the form

where and are entire functions with (see [1, page 64]). A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain. An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere. For example, the gamma function is meromorphic in the whole complex plane .

In the present paper, we initiate the study of functions which are meromorphic in the punctured disk with a Laurent expansion about the origin; see [2].

Let be the class of analytic functions with , which are convex and univalent in the open unit disk and for which

For functions and analytic in , we say that is subordinate to and write

if there exists an analytic function in such that

Furthermore, if the function is univalent in , then

This paper is divided into two sections; the first introduces a new class of meromorphically analytic functions, which is defined by means of a Hadamard product (or convolution) involving linear operator. The second section highlights some applications of the main results involving generalized hypergeometric functions.

2. Preliminaries

Let denote the class of meromorphic functions normalized by

which are analytic in the punctured unit disk . For , we denote by and the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order and convex of order in .

For functions    defined by

we denote the Hadamard product (or convolution) of and by

Cho et al. [3] and Ghanim and Darus [4] studied the following function:

Corresponding to the function and using the Hadamard product for , we define a new linear operator on by

The Hadamard product or convolution of the functions given by (10) with the functions and given, respectively, by

can be expressed as follows:

By applying the subordination definition, we introduce here a new class of meromorphic functions, which is defined as follows.

Definition 1. A function of the form (6) is said to be in the class if it satisfies the following subordination property:

where , , with condition and .

As for the second result of this paper on applications involving generalized hypergeometric functions, we need to utilize the well-known Gaussian hypergeometric function. One denotes the class of the function given by

for , and , where is the Pochhammer symbol. We note that

where

is the well-known Gaussian hypergeometric function.

Corresponding to the functions and given in (9) and using the Hadamard product for , we define a new linear operator on by

The meromorphic functions with the generalized hypergeometric functions were considered recently by Cho and Kim [5], Dziok and Srivastava [6, 7], Ghanim [8], Ghanim et al. [9, 10], and Liu and Srivastava [11, 12].

Now, it follows from (17) that

The subordination relation (13) in conjunction with (17) takes the following form:

Definition 2. A function of the form (6) is said to be in the class if it satisfies the subordination relation (19) above.

3. Characterization and Other Related Properties

In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a function of the form (6) to belong to the class of meromorphically analytic functions.

Theorem 3. The function is said to be a member of the class if and only if it satisfies

The equality is attained for the function given by

Proof. Let of the form (6) belong to the class . Then, in view of (12), we find that

Putting    and noting the fact that the denominator in the above inequality remains positive by virtue of the constraints stated in (13) for all , we easily arrive at the desired inequality (20) by letting .

Conversely, if we assume that the inequality (20) holds true in the simplified form (22), it can readily be shown that

which is equivalent to our condition of theorem, so that , hence the theorem.

Theorem 3 immediately yields the following result.

Corollary 4. If the function belongs to the class , then

where the equality holds true for the functions given by (21).

We now state the following growth and distortion properties for the class .

Theorem 5. If the function defined by (6) is in the class , then, for , one has

Proof. Since , Theorem 3 readily yields the inequality

Thus, for and utilizing (26), we have

Also from Theorem 3, we get

Hence

This completes the proof of Theorem 5.

We next determine the radii of meromorphic starlikeness and meromorphic convexity of the class , which are given by Theorems 6 and 7 below.

Theorem 6. If the function defined by (6) is in the class , then is meromorphic starlike of order in the disk , where

The equality is attained for the function given by (21).

Proof. It suffices to prove that

For , we have

Hence (32) holds true for

or

With the aid of (20) and (34), it is true to say that for fixed

Solving (35) for , we obtain

This completes the proof of Theorem 6.

Theorem 7. If the function defined by (6) is in the class , then is meromorphic convex of order in the disk , where

The equality is attained for the function given by (21).

Proof. By using the same technique employed in the proof of Theorem 6, we can show that

For and with the aid of Theorem 3, we have the assertion of Theorem 7.

4. Applications Involving Generalized Hypergeometric Functions

Theorem 8. The function is said to be a member of the class if and only if it satisfies

The equality is attained for the function given by

Proof. By using the same technique employed in the proof of Theorem 3 along with Definition 2, we can prove Theorem 8.

The following consequences of Theorem 8 can be deduced by applying (39) and (40) along with Definition 2.

Corollary 9. If the function belongs to the class , then

where the equality holds true for the functions given by (40).

Corollary 10. If the function defined by (6) is in the class , then is meromorphic starlike of order in the disk , where

The equality is attained for the function given by (40).

Corollary 11. If the function defined by (6) is in the class , then is meromorphic convex of order in the disk , where

The equality is attained for the function given by (40).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors read and approved the final paper.

Acknowledgment

The work here was fully supported by FRGSTOPDOWN/2013/ST06/UKM/01/1.

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