# 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 .

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.

- (ii)

- (iii)

- (iv)

## 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]).

The equality holds if and only if and .

Theorem 2.3.

then defined in Definition 1.1 is univalent in .

Proof. .

Remark 2.4.

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

Corollary 2.5.

Theorem 2.6.

then defined in Definition 1.1 is univalent in .

Proof. .

Remark 2.7.

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

Corollary 2.8.

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

## Authors’ Affiliations

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## Copyright

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