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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 267972, 7 pages
http://dx.doi.org/10.1155/2012/267972
Research Article

Some Convexity Properties of Certain General Integral Operators

1Department of Mathematics, University of Piteşti, Targul din Vale Street No. 1, 110040 Piteşti, Romania
2Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, Nicolae Iorga Street No. 11-13, 510009 Alba Iulia, Romania

Received 20 September 2011; Accepted 8 November 2011

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Vasile Marius Macarie 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.

Abstract

The main object of the present paper is to discuss some extensions of certain integral operators and to obtain their order of convexity. Several other closely related results are also considered.

1. Introduction

Let be the class of analytic functions defined in the open unit disk of the complex plane .

We denote by the subclass of consisting of all univalent functions in . A function is starlike function of order if it satisfies for some . We denote by the subclass of consisting of the functions which are starlike of order in . For , we obtain the class of starlike functions, denoted by .

A function is convex of order if it satisfies for some (). We denote by the subclass of consisting of the functions which are convex of order in . For , we obtain the class of convex functions, denoted by .

A function is in the class if

Frasin and Jahangiri introduced in [1] the family consisting of functions satisfying the condition For we have , and for we have .

In this paper, we will obtain the order of convexity of the following general integral operators: where the functions are in for all .

In order to prove our main results, we recall the following lemma.

Lemma 1.1 (see [2, General Schwarz Lemma]). 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. Main Results

Theorem 2.1. Let be in the class for all . If () for all , then the integral operator is in , where and .

Proof. Let be in the class . We have from (1.5) that Also Then and, hence, Applying the General Schwarz lemma, we have , for all . Therefore, from (2.6), we obtain From (1.4) and (2.7), we see that

Letting and for all in Theorem 2.1, we have the following corollary.

Corollary 2.2. Let be in the class for all . Then the integral operator defined in (1.5) is in , where and .

Letting and for all in Theorem 2.1, we have the following corollary.

Corollary 2.3. Let be in the class for all . If () for all , then the integral operator defined in (1.5) is in , where and .

Letting , , and for all in Theorem 2.1, we have the following corollary.

Corollary 2.4. Let be starlike functions in for all . If () for all , then the integral operator defined in (1.5) is convex in , where .

Theorem 2.5. Let be in the class for all . If () for all , then the integral operator is in , where and for all .

Proof. Let be in the class . It follows from (1.6) that and, hence, Applying the General Schwarz lemma, we have for all . Therefore, from (2.14), we obtain From (1.4) and (2.15), we see that This completes the proof.

Letting and for all in Theorem 2.5, we have the following corollary.

Corollary 2.6. Let be in the class for all . If () for all , then the integral operator defined in (1.6) is convex function in , where

Letting and for all in Theorem 2.5, we have the following corollary.

Corollary 2.7. Let be in the class for all . If () for all , then the integral operator defined in (1.6) is in , where and for all .

Letting , and for all in Theorem 2.5, we have the following corollary.

Corollary 2.8. Let be a starlike function in . If (), then the integral operator is convex in , where .

Theorem 2.9. Let be in the class for all . If () for all , then the integral operator is in , where and .

Proof. Let be in the class . It follows from (1.7) that So, from (2.21), we have Applying the General Schwarz lemma, we have for all . Therefore, from (2.22), we obtain From (1.4) and (2.23), we see that This completes the proof.

Letting and for all in Theorem 2.9, we have the following corollary.

Corollary 2.10. Let be in the class for all . If () for all , then the integral operator defined in (1.7) is convex function in , where

Letting and for all in Theorem 2.9, we have the following corollary.

Corollary 2.11. Let be in the class for all . If () for all , then the integral operator defined in (1.7) is in , where and .

Letting , , and for all in Theorem 2.9, we have the following corollary.

Corollary 2.12. Let be a starlike function in . If (), then the integral operator is convex in , where .

References

  1. B. A. Frasin and J. M. Jahangiri, “A new and comprehensive class of analytic functions,” Analele Universit\u a\c tii din Oradea. Fascicola Matematica, vol. 15, pp. 59–62, 2008. View at Zentralblatt MATH
  2. Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.