• Research Article
• Open Access

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

Journal of Inequalities and Applications20082008:957042

https://doi.org/10.1155/2008/957042

• Received: 10 June 2008
• Accepted: 21 July 2008
• Published:

## Abstract

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

## Keywords

• Analytic Function
• Differential Operator
• Integral Operator
• Unit Disk
• Schwarz Lemma

## 1. Introduction

Let denote the class of all functions of the form

which are analytic in the open unit disk , and .

For , Al-Oboudi  introduced the following operator:
If is given by (1.1), then from (1.3) and (1.4) we see that

with .

When , we get Sălăgean's differential operator .

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

Definition 1.1.

Let and , . We define the integral operator ,

where and is the Al-Oboudi differential operator.

Remark 1.2.
1. (i)
For , , , , and , we have Alexander integral operator

which was introduced in .
1. (ii)
For , , , , and , we have the integral operator

that was studied in .
1. (iii)
For , , , , , we have the integral operator

which was studied in .
1. (iv)
For , , , and , we have the integral operator

which was studied in [6, 7].

## 2. Main Results

The following lemmas will be required in our investigation.

Lemma 2.1 (see ).

If the function is regular in the unit disk , , and

for all , then the function is univalent in .

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

Let the analytic function be regular in and let . If, in , , then

and .

The equality holds if and only if and .

Theorem 2.3.

Let , , and , . If

then defined in Definition 1.1 is univalent in .

Proof. .

Since , , by (1.5), we have

for all .

for . This equality implies that
By hypothesis, since , and since we have

Thus .

Remark 2.4.

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

Corollary 2.5.

Let , and , . If

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 , similar results are given by using the Ruscheweyh differential operator.

## Authors’ Affiliations

(1)
Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Izmit-Kocaeli, Turkey

## References 