Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 207410, 8 pages

http://dx.doi.org/10.1155/2012/207410

## On Univalence Criteria for a General Integral Operator

^{1}Department of Mathematics, University of Piteşti, Târgu din Vale Street, No. 1, 110040 Piteşti, Romania^{2}Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, Nicolae Iorga Street, No. 11–13, 510009 Alba Iulia, Romania

Received 5 February 2012; Revised 19 May 2012; Accepted 19 May 2012

Academic Editor: Gestur Ólafsson

Copyright © 2012 Vasile Marius Macarie and Daniel Breaz. 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

We consider a new general integral operator, and we give sufficient conditions for the univalence of this integral operator in the open unit disk of the complex plane. Several consequences of the main results are also shown.

#### 1. Introduction

Let be the unit disk of the complex plane and let be the class of functions of the form that are analytic in and satisfy the usual normalization conditions . We denote by the subclass of consisting of all univalent functions in and consider the class of functions that are analytic in and that satisfy and for all .

In the present paper we obtain sufficient conditions for the following general integral operator to be in the class (The univalent functions are of importance in geometric functions theory and may have some applications in fluid mechanics and physics): where are complex numbers, , the functions for all and for all , where are positive integers.

For proving our main results we need the following theorems.

Theorem 1.1 (see [1]). *Let be a complex number, let ; and let . If
**
for all , then for any complex number with , the function
**
is in the class S.*

Theorem 1.2 (see [2]). * If the function is regular in and in , then for all and , the following inequalities hold:
**
These are equalities if and only if , where and .*

*Remark 1.3 (see [2]). *For , from inequality (1.5) we have
and hence,
Writing and letting , we get
for all .

#### 2. Main Results

Theorem 2.1. *Let . For , let be a complex number, let , and let . For , let and . If
**
then for every complex number with , the integral operator given by (1.2) is in the class .*

* Proof. *Let us define the function:
with , for all and , for all , and, thus, we obtain
The function is regular in and . We have
From (2.1), (2.2), and (2.6) we obtain
and by (2.3), we have
Using (2.8), by Theorem 1.1, it results that the integral operator given by (1.2) is in the class .

Letting in Theorem 2.1, we have the following.

Corollary 2.2. *Let , let , and let . For , let be a complex number, let , and let . If
**
then for every complex number with , the integral operator
**
is in the class .*

Letting in Theorem 2.1, we have the following.

Corollary 2.3. *Let , let , let , and let be a complex number. For , let and let . If
**
then for every complex number with , the integral operator
**
is in the class .*

For and in Theorem 2.1, we have the following.

Corollary 2.4. *Let , let , and let . For , let be a complex number and let . For , let . If
**
then for every complex number with , the integral operator given by (1.2) is in the class .*

Theorem 2.5. *Let . For , let be a complex number, let , and let . For , let and with , and . If
**
then for every complex number with , the integral operator given by (1.2) is in the class .*

* Proof. *We define the function:
with , for all and , for all .

We consider the function
We have
From (2.14), (2.15), and (2.20) we obtain
From (2.16), (2.19), and (2.21) we obtain for all .

We have , and, using Remark 1.3 we get
From (2.19) and (2.22), we obtain
for all .

From (2.17) and (2.23), we have
So, applying Theorem 1.1, we obtain that the integral operator given by (1.2) is in the class .

Letting in Theorem 2.5, we have the following.

Corollary 2.6. *Let , let , and let with . For , let be a complex number, and , and . If
**
then for every complex number with , the integral operator
**
is in the class . *

Letting in Theorem 2.5, we have the following.

Corollary 2.7. *Let , , and let be a complex number. For , let and with . If
**
then for every complex number with , the integral operator
**
is in the class .*

For and in Theorem 2.5, we have the following.

Corollary 2.8. *Let , , . For , let be a complex number and . For , let with , and .**If
**
then for every complex number with , the integral operator given by (1.2) is in the class .*

#### References

- N. N. Pascu, “An improvement of Becker's univalence criterion,” in
*Proceedings of the Commemorative Session: Simion Stoilow*, pp. 43–48, University of Brasov, 1987. - Z. Nehari,
*Conformal Mapping*, McGraw-Hill Book, New York, NY, USA, 1952, Dover, 1975. View at Zentralblatt MATH