#### 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.