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  • 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
(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, Arslanbey Campus, 41285 Izmit-Kocaeli, Turkey

References

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