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Sufficient Conditions for Univalence of an Integral Operator Defined by AlOboudi Differential Operator
Journal of Inequalities and Applications volumeÂ 2008, ArticleÂ number:Â 957042 (2008)
Abstract
We investigate the univalence of an integral operator defined by AlOboudi differential operator.
1. Introduction
Let denote the class of all functions of the form
which are analytic in the open unit disk , and .
For , AlOboudi [1] introduced the following operator:
If is given by (1.1), then from (1.3) and (1.4) we see that
with .
When , we get SÄƒlÄƒgean's differential operator [2].
By using the AlOboudi differential operator, we introduce the following integral operator.
Definition 1.1.
Let and , . We define the integral operator ,
where and is the AlOboudi differential operator.
Remark 1.2.

(i)
For , , , , and , we have Alexander integral operator
(1.7)
which was introduced in [3].

(ii)
For , , , , and , we have the integral operator
(1.8)
that was studied in [4].

(iii)
For , , , , , we have the integral operator
(1.9)
which was studied in [5].

(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]).
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 .
On the other hand, we obtain
for . This equality implies that
or equivalently
By differentiating the above equality, we get
After some calculus, we obtain
By hypothesis, since , and since we have
So, we obtain
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. .
By (2.9), we get
This inequality implies that
By Schwarz lemma (Lemma 2.2), we have
or
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.
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Bulut, S. Sufficient Conditions for Univalence of an Integral Operator Defined by AlOboudi Differential Operator. J Inequal Appl 2008, 957042 (2008). https://doi.org/10.1155/2008/957042
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DOI: https://doi.org/10.1155/2008/957042