# Sufficient Conditions for Univalence of an Integral Operator Defined by Al-Oboudi Differential Operator

- Serap Bulut
^{1}Email author

**2008**:957042

**DOI: **10.1155/2008/957042

© Serap Bulut. 2008

**Received: **10 June 2008

**Accepted: **21 July 2008

**Published: **22 July 2008

## Abstract

We investigate the univalence of an integral operator defined by Al-Oboudi differential operator.

## 1. Introduction

which are analytic in the open unit disk , and .

with .

When , we get Sălăgean's differential operator [2].

By using the Al-Oboudi differential operator, we introduce the following integral operator.

Definition 1.1.

where and is the Al-Oboudi differential operator.

- (i)For , , , , and , we have Alexander integral operator(1.7)

- (ii)For , , , , and , we have the integral operator(1.8)

- (iii)For , , , , , we have the integral operator(1.9)

- (iv)For , , , and , we have the integral operator(1.10)

## 2. Main Results

The following lemmas will be required in our investigation.

Lemma 2.1 (see [8]).

for all , then the function is univalent in .

Lemma 2.2 (Schwarz Lemma 2.2) (see [9, page 166]).

and .

The equality holds if and only if and .

Theorem 2.3.

then defined in Definition 1.1 is univalent in .

Proof. .

for all .

Thus .

Remark 2.4.

For , , , we have [5, Theorem 1].

Corollary 2.5.

and , then .

Theorem 2.6.

Let , and , . If

(i) ,

(ii) , and

(iii) ,

then defined in Definition 1.1 is univalent in .

Proof. .

for all .

So, by Lemma 2.1, .

Remark 2.7.

For , , , , , we have [7, Theorem 1].

Corollary 2.8.

Let , and , . If

(i) ,

(ii) , and

(iii) ,

then .

In [10], similar results are given by using the Ruscheweyh differential operator.

## Authors’ Affiliations

## References

- Al-Oboudi FM:
**On univalent functions defined by a generalized Sălăgean operator.***International Journal of Mathematics and Mathematical Sciences*2004,**2004**(27):1429–1436. 10.1155/S0161171204108090MATHMathSciNetView ArticleGoogle Scholar - Sălăgean GŞ:
**Subclasses of univalent functions.**In*Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), Lecture Notes in Mathematics*.*Volume 1013*. Springer, Berlin, Germany; 1983:362–372. 10.1007/BFb0066543Google Scholar - Alexander JW:
**Functions which map the interior of the unit circle upon simple regions.***Annals of Mathematics*1915,**17**(1):12–22. 10.2307/2007212MATHMathSciNetView ArticleGoogle Scholar - Miller SS, Mocanu PT, Reade MO:
**Starlike integral operators.***Pacific Journal of Mathematics*1978,**79**(1):157–168.MathSciNetView ArticleMATHGoogle Scholar - Breaz D, Breaz N:
**Two integral operators.***Studia Universitatis Babeş-Bolyai, Mathematica*2002,**47**(3):13–19.MATHMathSciNetGoogle Scholar - Pescar V:
**On some integral operations which preserve the univalence.***The Punjab University. Journal of Mathematics*1997,**30:**1–10.MATHMathSciNetGoogle Scholar - Pescar V, Owa S:
**Sufficient conditions for univalence of certain integral operators.***Indian Journal of Mathematics*2000,**42**(3):347–351.MATHMathSciNetGoogle Scholar - Becker J:
**Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen.***Journal für die Reine und Angewandte Mathematik*1972,**255:**23–43.MATHGoogle Scholar - Nehari Z:
*Conformal Mapping*. Dover, New York, NY, USA; 1975:vii+396.Google Scholar - Oros GI, Oros G, Breaz D:
**Sufficient conditions for univalence of an integral operator.***Journal of Inequalities and Applications*2008,**2008:**-7.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.