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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 721932, 4 pages
http://dx.doi.org/10.1155/2013/721932
Research Article

Kudriasov Type Univalence Criteria for Some Integral Operators

1Department of Mathematics and Computer Science, Faculty of Mathematics and Computer Science, Transilvania University of Braşov, 500091 Braşov, Romania
2Department of Mathematics and Computer Science, Faculty of Science, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba, Romania

Received 30 September 2013; Accepted 4 December 2013

Academic Editor: Mohamed Kamal Aouf

Copyright © 2013 Virgil Pescar and Nicoleta 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.

Abstract

We consider some integral operators defined by analytic functions in the open unit disk and derive new univalence criteria for these operators, using Kudriasov condition for a function to be univalent.

1. Introduction

Let be the class of functions which are analytic in the open unit disk of the form normalized by .

We denote by the subclass of consisting of functions , which are univalent in .

We consider the integral operators for , being complex numbers, , , and the functions , .

Some univalence criteria for these integral operators were studied in [1]. Applying univalence conditions given by Kudriasov [2] and Pascu [3], we obtain new Kudriasov type univalence criteria for these two integral operators.

2. Preliminaries

Various generalizations of Becker’s univalence criteria for analytic functions given in [4] were obtained by many authors. For example, the result obtained by Pascu in [3] is also known as an improvement of Becker’s univalence criteria. This result or other similar generalizations of Becker’s univalence criteria have been used further to derive new univalence criteria for integral operators (see, e.g., some relatively recent works as [1, 5, 6]). In this paper we use Pascu improvement of Becker’s univalence criteria and also another univalence condition for a function to be univalent, given by Kudriasov in [2]. There are also some papers devoted to univalence criteria that use some Kudriasov type conditions (see, e.g., the work [1] containing a chapter dedicated to Kudriasov type univalence conditions and other papers as, e.g., [710]).

The following univalence criteria are given by Kudriasov for a regular function.

Lemma 1 (see [2]). Let be a regular function in , . If for all , where , the function is univalent in .

Remark 2. The constant is a solution of the equation . An approximation of this solution in MATLAB environment is 3.03902118847875. However, we call this constant in our further results like Kudriasov gave it, approximately equal to 3.05.

The improvement of Becker’s univalence condition is given by Pascu for integral operators as follows.

Lemma 3 (see [3]). Let be a complex number, , and the function . If for all , then for every complex number , , the function is regular and univalent in .

3. Main Results

Theorem 4. Let , be complex numbers, , , the functions , , , , and the positive real number .
If then , , and for every complex number , , the integral operator is in the class .

Proof. Let us consider the function
The function is regular in and . We have for all .
From (9), we obtain further
By (6) and Lemma 1, we have , , and hence we obtain
From (10) and (11), we get for all .
Now we consider the following cases.
(1) . The function , , , , is increasing and we obtain
From (12) and (13), we obtain for all .
Using the hypothesis condition (7), from (14), we have for all .
(2) . We notice that the function is decreasing function, and we obtain
Using the last inequality in (12), we have for all .
Now using the hypothesis condition (7), from (18), we get for all .
Hence, based on the conditions obtained in (15) and in (19), applying Lemma 3, we have that .

Theorem 5. Let , be complex numbers, , , the functions , , , , and the positive real number, .
If then , , and for every complex number , , the integral operator is in the class .

Proof. By (20) and Lemma 1, we obtain that , .
We consider the function The function is regular in and .
We have for all .
Further we obtain
From (24) and from the Kudriasov condition within the hypothesis, (20), we have for all .
Let us consider the function , , , . We have
By (25), (26), and (21) we obtain for all .
Now from (27) and Lemma 3, it results that .

4. Corollaries

Corollary 1. Let be complex number, , the functions , , , and the positive real number, .
If then , , and the integral operator defined by is in the class .

Proof. From (29), we have and for , , from Theorem 4, we obtain Corollary 1.

Corollary 2. Let , be complex numbers, , , the functions , , , , and the positive real number .
If then , and the integral operator defined by belongs to the class .

Corollary 3. Let , be complex numbers, , , the functions , , , , and the positive real number, .
If then , , and the integral operator defined by is in the class .

References

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