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

Journal of Inequalities and Applications20082008:957042

DOI: 10.1155/2008/957042

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

Let denote the class of all functions of the form
(1.1)

which are analytic in the open unit disk , and .

For , Al-Oboudi [1] introduced the following operator:
(1.2)
(1.3)
(1.4)
If is given by (1.1), then from (1.3) and (1.4) we see that
(1.5)

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.

Let and , . We define the integral operator ,
(1.6)

where and is the Al-Oboudi differential operator.

Remark 1.2.
  1. (i)
    For , , , , and , we have Alexander integral operator
    (1.7)
     
which was introduced in [3].
  1. (ii)
    For , , , , and , we have the integral operator
    (1.8)
     
that was studied in [4].
  1. (iii)
    For , , , , , we have the integral operator
    (1.9)
     
which was studied in [5].
  1. (iv)
    For , , , and , we have the integral operator
    (1.10)
     

which was studied in [6, 7].

2. Main Results

The following lemmas will be required in our investigation.

Lemma 2.1 (see [8]).

If the function is regular in the unit disk , , and
(2.1)

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
(2.2)

and .

The equality holds if and only if and .

Theorem 2.3.

Let , , and , . If
(2.3)

then defined in Definition 1.1 is univalent in .

Proof. .

Since , , by (1.5), we have
(2.4)

for all .

On the other hand, we obtain
(2.5)
for . This equality implies that
(2.6)
or equivalently
(2.7)
By differentiating the above equality, we get
(2.8)
After some calculus, we obtain
(2.9)
By hypothesis, since , and since we have
(2.10)
So, we obtain
(2.11)

Thus .

Remark 2.4.

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

Corollary 2.5.

Let , and , . If
(2.12)

and , then .

Theorem 2.6.

Let , and , . If

(i) ,

(ii) , and

(iii) ,

then defined in Definition 1.1 is univalent in .

Proof. .

By (2.9), we get
(2.13)
This inequality implies that
(2.14)
By Schwarz lemma (Lemma 2.2), we have
(2.15)
or
(2.16)

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

(1)
Civil Aviation College, Kocaeli University

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Copyright

© Serap Bulut. 2008

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.