International Journal of Mathematics and Mathematical Sciences

Volume 2013 (2013), Article ID 769537, 5 pages

http://dx.doi.org/10.1155/2013/769537

Research Article

Univalence of a New General Integral Operator Associated with the -Hypergeometric Function

Huda Aldweby and Maslina Darus

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

Received 11 December 2012; Revised 3 February 2013; Accepted 17 February 2013

Academic Editor: Shyam Kalla

Copyright © 2013 Huda Aldweby 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.

Motivated by the familiar -hypergeometric functions, we introduce a new family of integral operators and obtain new sufficient conditions of univalence criteria. Several corollaries and consequences of the main results are also pointed out.

1. Introduction

Let denote the class of functions of the form

which are analytic in the open unit disk , and the class of functions which are univalent in .

Let , where is defined by (1) and is given by

Then the Hadamard product (or convolution) of the functions and is defined by

For complex parameters and, we define the -hypergeometric function by

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

By using the ratio test, we should note that, if , the series (4) converges absolutely for if . For more mathematical background of these functions, one may refer to [1].

Corresponding to the function defined by (4), consider

Recently, the authors [2] defined the linear operator by

where

It should be remarked that the linear operator (7) is a generalization of many operators considered earlier. For , and , we obtain the Dziok-Srivastava linear operator [3] (for), so that it includes (as its special cases) various other linear operators introduced and studied by Ruscheweyh [4], Carlson and Shaffer [5] and the Bernardi-Libera-Livingston operators [6–8].

The -difference operator is defined by

where is the ordinary derivative. For more properties of see [9, 10].

Lemma 1 (see [2]). Let ; then

(i) for , and , one has . (ii) For , and , one has and , where is the -derivative defined by (9).

Definition 2. A function is said to be in the class if it is satisfying the condition

where is the operator defined by (7).

Note that , where the class of analytic and univalent functions was introduced and studied by Frasin and Darus [11].

Using the operator , we now introduce the following new general integral operator.

For , , and , we define the integral operator by

where .

Remark 3. It is interesting to note that the integral operator generalizes many operators introduced and studied by several authors, for example,

(1) for , and , where and , we obtain the following integral operator introduced and studied by Selvaraj and Karthikeyan [12]:

where for convenience , and is the Dziok-Srivastava operator [3].

(2) For , and , we obtain the integral operator

studied recently by Breaz et al. [13].

(3) For , and , we obtain the integral operator

introduced and studied by D. Breaz and N. Breaz [14].

(4) For , and , we obtain the integral operator

introduced by Selvaraj and Karthikeyan [12].

(5) For , and , we obtain the integral operator

recently introduced and studied by Breaz and Güney [15].

(6) For , and , where and , we obtain the integral operator

introduced and studied by Pescar [16].

In order to derive our main results, we have to recall the following univalence criteria.

Lemma 4 (see [17, 18]). Let with . If satisfies

for all , then the integral operator

is in the class .

Lemma 5 (see [16]). Let with , with . If satisfies

for all then the integral operator

is in the class .

Lemma 6 (Generalized Schwarz Lemma, see [19]). (Generalized Schwarz Lemma) Let the function be analytic in the disk , with for fixed . If has one zero with multiplicity order bigger that for , then

Equality can hold only if

where is constant.

2. Univalence Conditions for

Theorem 7. Let for all , and with

If for all , and

then the integral operator defined by (11) is analytic and univalent in .

Proof. From the definition of the operator it can be observed that

and for , we have

We define the function by the form

Therefore

Differentiating logarithmically and multiplying by on both sides of (29)

Thus we have

So

Since , and for all , then from the Schwarz Lemma and (10), we obtain

which, in the light of the hypothesis (24), yields

Applying Lemma (1) for the function we obtain that is univalent.

Taking , and in Theorem 7, we have the following.

Corollary 8 (see [12]). Let for all and with

If

and for all , then the integral operator defined by (12) is analytic and univalent in .

Taking (for all),, and in Theorem 7, we have the following.

Corollary 9. Let for all and with

If

and for all , then the integral operator defined by (13) is analytic and univalent in .

Theorem 10. Let for all , and with

If for all , and

then the integral operator defined by (11) is analytic and univalent in .

Proof. From the proof of Theorem 7, we have

Thus we have

From this result and using the proof of Theorem 7 we obtain

Since, then we have

Applying Lemma (4) for the function we obtain that is univalent.

Taking (for all),, and in Theorem 10, we have the following.

Corollary 11. Let for all ; , and with

If for all

then the integral operator defined by (13) is analytic and univalent in .

Letting and in Corollary 11, we have the following.

Corollary 12. Let , and with

If

then the integral operator defined by (17) is analytic and univalent in .

Remark 13. Many other interesting corollaries and results can be obtained by specializing the parameters in Theorem 10; for example, see [13, 20, 21].

Acknowledgments

The work presented here was partially supported by GUP-2012-023 and UKM-DLP-2011-050.

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