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

  • Serap Bulut
Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window denote the class of all functions of the form

which are analytic in the open unit disk Open image in new window , and Open image in new window .

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

with Open image in new window .

When Open image in new window , we get Sălăgean's differential operator [2].

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

Definition 1.1.

where Open image in new window and Open image in new window is the Al-Oboudi differential operator.

which was studied in [6, 7].

2. Main Results

The following lemmas will be required in our investigation.

Lemma 2.1 (see [8]).

for all Open image in new window , then the function Open image in new window is univalent in Open image in new window .

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

and Open image in new window .

The equality holds if and only if Open image in new window and Open image in new window .

Theorem 2.3.

then Open image in new window defined in Definition 1.1 is univalent in Open image in new window .

Proof. .

for all Open image in new window .

On the other hand, we obtain
for Open image in new window . This equality implies that
or equivalently
By differentiating the above equality, we get
After some calculus, we obtain
By hypothesis, since Open image in new window , and since Open image in new window we have
So, we obtain

Thus Open image in new window .

Remark 2.4.

For Open image in new window , Open image in new window , Open image in new window , we have [5, Theorem 1].

Corollary 2.5.

and Open image in new window , then Open image in new window .

Theorem 2.6.

Let Open image in new window , Open image in new window and Open image in new window , Open image in new window . If

(i) Open image in new window ,

(ii) Open image in new window , and

(iii) Open image in new window ,

then Open image in new window defined in Definition 1.1 is univalent in Open image in new window .

Proof. .

By (2.9), we get
This inequality implies that
By Schwarz lemma (Lemma 2.2), we have

for all Open image in new window .

So, by Lemma 2.1, Open image in new window .

Remark 2.7.

For Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , we have [7, Theorem 1].

Corollary 2.8.

Let Open image in new window , Open image in new window and Open image in new window , Open image in new window . If

(i) Open image in new window ,

(ii) Open image in new window , and

(iii) Open image in new window ,

then Open image in new window .

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

References

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Copyright information

© 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.

Authors and Affiliations

  1. 1.Civil Aviation CollegeKocaeli UniversityIzmit-KocaeliTurkey

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