Journal of Applied Mathematics

Volume 2014 (2014), Article ID 134910, 8 pages

http://dx.doi.org/10.1155/2014/134910

Research Article

Some Properties for an Integral Operator Defined by Generalized Hypergeometric Function

Ibtisam Aldawish and Maslina Darus

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

Received 10 May 2014; Accepted 4 July 2014; Published 4 August 2014

Academic Editor: Song Cen

Copyright © 2014 Ibtisam Aldawish 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.

We define a new general integral operator for meromorphic functions involving the generalized hypergeometric function. Furthermore, we study the characterization and other properties for this operator.

1. Introduction

Let denote the class of functions of the form

which are analytic in the punctured open unit disk .

We say that a function is meromorphic starlike of order and belongs to the class , if it satisfies the following inequality:

A function is a meromorphic convex function of order , if satisfies the following inequality:

and we denote this class by .

Also, let be the subclass of , consisting of the functions , which satisfy the inequality

where the subclass was introduced and studied by Wang et al. [1] and Nehari and Netanyahu [2].

A -hypergeometric function is a power series in one complex variable with power series coefficients which depend (apart from ) on complex upper parameters , , as follows:

with , where when ,  , and is the Pochhammer symbol defined by

Tracing back the history of basic hypergeometric series defined by (5) has brought us to Heine dated in 1846. Therefore it is sometimes called Heine’s series. For brief survey on -hypergeometric functions, one may refer to [3–5] (see also [6–8]).

For , , and , the basic hypergeometric function defined in (5) takes the form

which converges absolutely in the open unit disk .

Corresponding to the function , let

Analogous to the differential operator defined in [9] which involves the -hypergeometric functions on the normalized analytic functions, we define the following differential operator on the space of meromorphic functions in the class by

If , then from (9) we may deduce that

where

, and .

It should be remarked that the operator given by (10) is a generalization of many other operators considered earlier; for example, consider the following.

(i)For , , , , the operator was defined by Frasin and Darus [10] and studied by El-Ashwah and Aouf [11].(ii)For , , , , ,  , the operator was imposed by Liu and Srivastava [12].(iii)For , , , , , the operator was introduced and studied by Liu and Srivastava [13]. Further, we note in passing that this operator is closely related to the Carlson-Shaffer operator defined on the space of analytic univalent functions in .(iv)For , , , , , the operator , where is the differential operator, was introduced by Ganigi and Uralegaddi [14], and then it was generalized by Yang [15].

Now, making use of the differential operator given by (10), we introduce the following integral operator in the class .

Definition 1. Let , . We define the integral operator by

Remark 2. Note that for , , , , we obtain the integral operator defined by Mohammed and Darus [16]; see also [17–19].

With the aid of differential operator given by (10), we define a new subclass of functions in as follows.

Definition 3. Let a function be analytic in . Then is in the class if, and only if, satisfies

where is defined in (10) and .

Definition 4. Let a function be analytic in . Then is in the class if, and only if, satisfies

where is defined in (10) and , .

Definition 5. Let a function be analytic in . Then is in the class if, and only if, satisfies

where is defined in (10) and .

Definition 6. Let a function be analytic in . Then is in the class if, and only if, satisfies

where is defined in (10) and .

Remark 7. Putting , , and in Definition 3, we obtain the result corresponding to the class mentioned in (3).

For , , and in Definitions 4, 5, and 6, we obtain and the classes of meromorphic functions, introduced and studied by Mohammed and Darus [20].

2. Preliminary Definitions

We begin by recalling each of the following definitions of subclasses of meromorphic functions , and which will be required in our investigation.

Definition 8 (see [20]). Let a function be analytic in . Then is in the class if, and only if, satisfies

where , .

Definition 9 (see [20]). Let a function be analytic in . Then is in the class if, and only if, satisfies

where .

Definition 10 (see [20]). Let a function be analytic in . Then is in the class if, and only if, satisfies

where .

3. Main Results

In this section, we study some properties for the integral operator defined by (12) of the subclasses given by Definitions 3, 4, 5, and 6.

Theorem 11. Let ,  for and . If

and , then is in the class , .

Proof. By differentiating the equality given by (12), we get

Again, differentiating (21), we deduce that

By using (21) and (23), we have

Multiplying (24) by yields that

or, equivalently, that

Thus, we get

That is,

Taking the real parts of both terms of the last expression and using the fact that for all , we impose

Setting

and since

we conclude that

Hence, by (20), we have , so that , which evidently completes the proof of Theorem 11.

Theorem 12. Let , for , and . If

then is in the class , where .

Proof. We know from the proof of Theorem 11 that

which is equivalent to

Taking the real part of both terms of (35), we get

Now, since , we have

Letting

and in view of (33), it follows that , so that . This completes the proof of Theorem 12.

Theorem 13. Let , for , and . If

then is in the class .

Proof. We want to show that

We find from (26) and (36) that

Since , it follows from Definition 5 that

From hypothesis (39), we note that

Hence . Therefore, we complete the proof of Theorem 13.

Theorem 14. Let for , and . If

then is in the class .

Proof. Our aim is to prove that

With the aid of (26), (36) can be written as follows:

Since , it follows from Definition 6 and by a simple calculation that

which means that , and we obtain the proof of the theorem.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Ibtisam Aldawish and Maslina Darus read and approved the final paper.

Acknowledgment

The work presented here was partially supported by UKM’s Grant FRGSTOPDOWN/2013/ST06/UKM/01/1.

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