International Journal of Mathematics and Mathematical Sciences

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.