Research Article | Open Access

Serap Bulut, Daniel Breaz, "Univalency and Convexity Conditions for a General Integral Operator", *Chinese Journal of Mathematics*, vol. 2014, Article ID 923984, 4 pages, 2014. https://doi.org/10.1155/2014/923984

# Univalency and Convexity Conditions for a General Integral Operator

**Academic Editor:**H. You

#### Abstract

For analytic functions and in the open unit disc , a new general integral operator is introduced. The main objective of this paper is to obtain univalence condition and order of convexity for this general integral operator.

#### 1. Introduction and Preliminaries

Let be the class of all functions of the form which are analytic in the open unit disk Also let denote the subclass of consisting of functions which are univalent in .

A function is said to be starlike of order if it satisfies the inequality for all . We say that is in the class for such functions.

A function is said to be convex of order if it satisfies the inequality for all . We say that is in the class for such functions.

We note that if and only if .

A function belongs to the class if it satisfies the inequality for all .

The family which contains the functions that satisfy the condition was studied by Frasin and Jahangiri [1].

*Remark 1. *This family is a comprehensive class of analytic functions that contains other new classes of analytic univalent functions as well as some very well-known ones. For example,(i)for , we have the class
(ii)for , we have the class
(iii)for , the class
introduced by Frasin and Darus [2].

In this paper, we introduce a new general integral operator defined by where , , and for all .

*Remark 2. *For , , , and , we have the integral operator
introduced by Ularu and Breaz [3].

The following results will be required in our investigation.

General Schwarz Lemma (see [4]). *Let the function ** be regular in the disk **, with ** for fixed **. If ** has one zero with multiplicity order bigger than ** for **, then **The equality can hold only if **, where ** is constant. *

Theorem A (see [5]). *If the function is regular in the unit disk , and
**
for all , then the function is univalent in .*

Theorem B (see [6]). *If satisfies the condition
**
then is univalent in .*

#### 2. Main Results

Theorem 3. *Let , where satisfies the condition
**, and for all . If
**
for all , then the integral operator defined by (10) is in the univalent function class .*

*Proof. *From (10), we obtain
for . This equality implies that
By differentiating the above equality, we get
or equivalently
So we find
From the hypothesis, we have ; then by the general Schwarz lemma, we obtain that
Thus we have
Let us define the function , , . Then we obtain
for all . It follows from (26) that

Hence from (18), (25), and (27) we find
By applying Theorem A for the function , we prove that is in the univalent function class .

Setting , , , and in Theorem 3, we obtain the following consequence of Theorem 3.

Corollary 4 (see [3]). *Let , where satisfies the condition**, and . If
**
then the integral operator defined by (11) is in the univalent function class .*

Theorem 5. *Let , where , , and for all . If
**
for all , and if
**
then the integral operator defined by (10) is in the class , where
*

*Proof. *From (22), we have
It follows that
Since , for all , applying general Schwarz lemma and using (35), we obtain
This implies that the integral operator .

Setting in Theorem 5, we have the following.

Corollary 6. *Let , where , , and for all . If
**
for all , and if
**
then the integral operator defined by (10) is in the class , where
*

Setting in Theorem 5, we have the following.

Corollary 7. *Let , where , , and for all . If
**
for all , and if
**
then the integral operator defined by (10) is in the class , where
*

Setting , , , and in Theorem 5, we obtain the following consequence of Theorem 5.

Corollary 8 (see [3]). *Let , where , , and . If
**
then the integral operator defined by (11) is in the class , where
*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

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#### Copyright

Copyright © 2014 Serap Bulut and Daniel Breaz. 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.