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

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.