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Journal of Mathematics
Volume 2013 (2013), Article ID 260127, 4 pages
http://dx.doi.org/10.1155/2013/260127
Research Article

General Integral Operator of Analytic Functions Involving Functions with Positive Real Part

Department of Mathematics, Faculty of Science, Al al-Bayt University, P.O. Box 130095, Mafraq, Jordan

Received 22 August 2012; Accepted 21 October 2012

Academic Editor: S. T. Ali

Copyright © 2013 B. A. Frasin. 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.

Abstract

Let be the integral operator defined by where each of the functions and are, respectively, analytic functions and functions with positive real part defined in the open unit disk for all . The object of this paper is to obtain several univalence conditions for this integral operator. Our main results contain some interesting corollaries as special cases.

1. Introduction and Definitions

Let denote the class of the normalized functions of the form which are analytic in the open unit disk . Further, by we shall denote the class of all functions in which are univalent in . Also, let be the class of all functions which are analytic in and satisfy .

Frasin and Darus [1] (see also [2]) defined the family so that it consists of functions satisfying the condition

Very recently many authors studied the problem of integral operators which preserve the class (see, e.g., [315]). In this paper, we obtain new sufficient conditions for the univalence of the general integral operator defined by where , , , and for all .

Here and throughout in the sequel, every many-valued function is taken with the principal branch.

Remark 1. Note that the integral operator generalizes the following operators introduced and studied by several authors:(1)If we let , for all , in (3), we obtain the integral operator: introduced and studied by D. Breaz and N. Breaz [16].(2)If we let , for all , in (3), we obtain the integral operator: introduced and studied by Frasin [17].(3)If we let and , for all , in (3), we obtain the integral operator: introduced and studied by D. Breaz and N. Breaz [16].

In order to derive our main results, we have to recall here the following lemmas.

Lemma 2 (see [18]). Let with If satisfies for all , then, for any complex number with , the integral operator is in the class .

Lemma 3 (see [13]). Let with with If satisfies

for all , then the integral operator defined by (8) is in the class .

Lemma 4 (see [19]). If , then we have

Lemma 5 (see [20]). If , then Also, we need the following general Schwarz Lemma.

Lemma 6 (see [21]). 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.

2. Univalence Conditions for the Operator

We first prove the following theorem.

Theorem 7. Let ; and for all . Let with . If then the integral operator defined by (3) is in the class .

Proof. Define the regular function by
Then it is easy to see that and . Differentiating both sides of (16) logarithmically, we obtain
Thus, we have
Since and for all , from (18), (11), and (10), we obtain
Multiply both sides of (19) by , we get for all .
Let us denote , , , and . It is easy to prove that
From (20), (21), and the hypothesis (14), we have
for all . Applying Lemma 2 for the function , we prove that .

Letting , , , , and in Theorem 7, we obtain the following corollary.

Corollary 8. Let ; and . Also, let with . If then the integral operator defined by is in the class .

If we set in Corollary 8, we have the following.

Corollary 9. Let and . Also, let with . If then the integral operator defined by (24) is in the class .

Next, we prove the following theorem.

Theorem 10. Let satisfies , and for all , where ,, with , then the integral operator defined by (3) is in the class .

Proof. Suppose that for all . Thus, we have where for all . Differentiating both sides of (27) logarithmically, we obtain
Define the regular function as in (15). Thus from (28) and (17), we have
Form the hypothesis (26) and (29), we immediately have for all . Applying Lemma 6, we obtain
Thus from (29) and (31) we have, for all . Let us denote , , , and . It is easy to prove that the maximum is attained at the point , and thus we have
In view of this inequality and (32), we obtain
Applying Lemma 2 for the function , we prove that .

Letting , , , and in Theorem 10, we have the following corollary.

Corollary 11. Let satisfies , and where , with , then the integral operator defined by (24) is in the class .

Using Lemma 3, we derive the following theorem.

Theorem 12. Suppose that each of the functions satisfies , and
for all , where ,, and , then the integral operator defined by (3) is in the class .

Proof. From (29), we have
Now by using the hypothesis (36), we obtain
Finally, by applying Lemma 3, we conclude that .

Letting , , , and in Theorem 12, we have the following corollary.

Corollary 13. Suppose that the functions satisfy , and where , , and , then the integral operator defined by (24) is in the class .

Acknowledgment

The author would like to thank the referee for his helpful comments and suggestions.

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