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International Journal of Mathematics and Mathematical Sciences

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

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

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

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.

#### References

- G. Gasper and M. Rahman,
*Basic Hypergeometric Series*, vol. 35 of*Encyclopedia of Mathematics and its Applications*, Cambridge University Press, Cambridge, UK, 1990. View at MathSciNet - A. Mohammed and M. Darus, “A generalized operator involving the $q$-hypergeometric function,”
*Mathematici Vesnik*, Available online 10.06.2012, 12 pages. View at Publisher · View at Google Scholar - J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,”
*Applied Mathematics and Computation*, vol. 103, no. 1, pp. 1–13, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - S. Ruscheweyh, “New criteria for univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 49, pp. 109–115, 1975. View at Google Scholar · View at MathSciNet - B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,”
*SIAM Journal on Mathematical Analysis*, vol. 15, no. 4, pp. 737–745, 1984. View at Publisher · View at Google Scholar · View at MathSciNet - S. D. Bernardi, “Convex and starlike univalent functions,”
*Transactions of the American Mathematical Society*, vol. 135, pp. 429–446, 1969. View at Google Scholar · View at MathSciNet - R. J. Libera, “Some classes of regular univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 16, pp. 755–758, 1965. View at Google Scholar · View at MathSciNet - A. E. Livingston, “On the radius of univalence of certain analytic functions,”
*Proceedings of the American Mathematical Society*, vol. 17, pp. 352–357, 1966. View at Google Scholar · View at MathSciNet - H. Exton,
*q-Hypergeometric Functions and Applications*, Ellis Horwood, Chichester, UK, 1983. View at MathSciNet - H. A. Ghany, “$q$-derivative of basic hypergeometric series with respect to parameters,”
*International Journal of Mathematical Analysis*, vol. 3, no. 33-36, pp. 1617–1632, 2009. View at Google Scholar · View at MathSciNet - B. A. Frasin and M. Darus, “On certain analytic univalent functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 25, no. 5, pp. 305–310, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - C. Selvaraj and K. R. Karthikeyan, “Sufficient conditions for univalence of a general integral operator,”
*Acta Universitatis Apulensis*, no. 17, pp. 87–94, 2009. View at Google Scholar · View at MathSciNet - D. Breaz, N. Breaz, and H. M. Srivastava, “An extension of the univalent condition for a family of integral operators,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 41–44, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - D. Breaz and N. Breaz, “Two integral operators,”
*Studia Universitatis Babes-Bolyai, Mathematica, Cluj-Napoca*, vol. 47, no. 3, pp. 13–19, 2002. View at Google Scholar · View at MathSciNet - D. Breaz and H. Ö. Güney, “The integral operator on the classes ${S}_{\alpha}^{\ast}(b)$ and ${C}_{\alpha}(b)$,”
*Journal of Mathematical Inequalities*, vol. 2, no. 1, pp. 97–100, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - V. Pescar, “A new generalization of Ahlfors's and Becker's criterion of univalence,”
*Malaysian Mathematical Society Bulletin*, vol. 19, no. 2, pp. 53–54, 1996. View at Google Scholar · View at MathSciNet - N. N. Pascu, “On a univalence criterion. II,” in
*Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 1985)*, vol. 85, pp. 153–154, Babeş-Bolyai University, Cluj-Napoca, Romania, 1985. View at Google Scholar · View at MathSciNet - N. N. Pascu, “An improvement of Becker's univalence criterion,” in
*Proceedings of the Commemorative Session: Simion Stoïlow*, pp. 43–48, University of Braşov, Braşov, Romania, 1987. View at MathSciNet - Z. Nehari,
*Conformal Mapping*, Dover, New York, NY, USA, 1975. View at MathSciNet - D. Breaz and H. Ö. Güney, “On the univalence criterion of a general integral operator,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 702715, 8 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - G. I. Oros, G. Oros, and D. Breaz, “Sufficient conditions for univalence of an integral operator,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 127645, 7 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet