Chinese Journal of Mathematics

Chinese Journal of Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 923984 | 4 pages | https://doi.org/10.1155/2014/923984

Univalency and Convexity Conditions for a General Integral Operator

Academic Editor: H. You
Received24 Oct 2013
Accepted26 Dec 2013
Published20 Feb 2014

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

  1. B. A. Frasin and J. M. Jahangiri, “A new and comprehensive class of analytic functions,” Analele Universităţii din Oradea, vol. 15, pp. 59–62, 2008. View at: Google Scholar
  2. B. A. Frasin and M. Darus, “On certain analytic univalent functions,” International Journal of Mathematics and Mathematical Sciences, vol. 25, no. 5, pp. 305–310, 2001. View at: Publisher Site | Google Scholar
  3. N. Ularu and D. Breaz, “Univalence criterion and convexity for an integral operator,” Applied Mathematics Letters, vol. 25, no. 3, pp. 658–661, 2012. View at: Publisher Site | Google Scholar
  4. Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.
  5. J. Becker, “Löwnersche differentialgleichung und quasikonform fortsetzbare schlichte funktionen,” Journal für die Reine und Angewandte Mathematik, vol. 255, pp. 23–43, 1972. View at: Publisher Site | Google Scholar
  6. S. Ozaki and M. Nunokawa, “The Schwarzian derivative and univalent functions,” Proceedings of the American Mathematical Society, vol. 33, pp. 392–394, 1972. View at: Publisher Site | Google Scholar

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.


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