## Journal of Inequalities and Applications

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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

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

## References

1. Al-Oboudi FM: On univalent functions defined by a generalized Sălăgean operator. International Journal of Mathematics and Mathematical Sciences 2004,2004(27):1429–1436. 10.1155/S0161171204108090
2. Sălăgean GŞ: Subclasses of univalent functions. In Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), Lecture Notes in Mathematics. Volume 1013. Springer, Berlin, Germany; 1983:362–372. 10.1007/BFb0066543Google Scholar
3. Alexander JW: Functions which map the interior of the unit circle upon simple regions. Annals of Mathematics 1915,17(1):12–22. 10.2307/2007212
4. Miller SS, Mocanu PT, Reade MO: Starlike integral operators. Pacific Journal of Mathematics 1978,79(1):157–168.
5. Breaz D, Breaz N: Two integral operators. Studia Universitatis Babeş-Bolyai, Mathematica 2002,47(3):13–19.
6. Pescar V: On some integral operations which preserve the univalence. The Punjab University. Journal of Mathematics 1997, 30: 1–10.
7. Pescar V, Owa S: Sufficient conditions for univalence of certain integral operators. Indian Journal of Mathematics 2000,42(3):347–351.
8. Becker J: Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. Journal für die Reine und Angewandte Mathematik 1972, 255: 23–43.
9. Nehari Z: Conformal Mapping. Dover, New York, NY, USA; 1975:vii+396.Google Scholar
10. Oros GI, Oros G, Breaz D: Sufficient conditions for univalence of an integral operator. Journal of Inequalities and Applications 2008, 2008:-7.Google Scholar