About this Journal Submit a Manuscript Table of Contents
International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 103521, 10 pages
http://dx.doi.org/10.1155/2011/103521
Research Article

On Certain Subclasses of Analytic Functions Defined by Differential Subordination

Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

Received 3 June 2011; Accepted 25 August 2011

Academic Editor: Stanisława R. Kanas

Copyright © 2011 Hesam Mahzoon. 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 introduce and study certain subclasses of analytic functions which are defined by differential subordination. Coefficient inequalities, some properties of neighborhoods, distortion and covering theorems, radius of starlikeness, and convexity for these subclasses are given.

1. Introduction

Let be the class of analytic functions of the form defined in the open unit disc

Let be the class of functions analytic in such that , .

For any two functions and in , is said to be subordinate to that is denoted , if there exists an analytic function such that [1].

Definition 1.1 (see [2]). For and , the Al-Oboudi operator is defined as , , and .

For , we get Sǎlǎgean differential operator [3].

Further, if , then For any function and , the -neighborhood of is defined as In particular, for the identity function , we see that The concept of neighborhoods was first introduced by Goodman [4] and then generalized by Ruscheweyh [5].

Definition 1.2. A function is said to be in the class if where , , , and .

We observe that   [6],    [7], the class of starlike functions of order and   [7], the class of convex functions of order .

2. Neighborhoods for the Class

Theorem 2.1. A function belongs to the class if and only if for , , , and .

Proof. Let . Then, Therefore, Hence, Thus, Taking , for sufficiently small with , the denominator of (2.5) is positive and so it is positive for all with , since is analytic for . Then, inequality (2.5) yields Equivalently, and (2.1) follows upon letting .
Conversely, for , , we have . That is, From (2.1), we have This proves that and hence .

Theorem 2.2. If then .

Proof. It follows from (2.1) that if , then which implies Using (1.4), we get the result.

3. Neighborhoods for the Classes and

Definition 3.1. A function is said to be in the class if it satisfies where , and .

Definition 3.2. A function is said to be in the class if it satisfies where , and .

Lemma 3.3. A function belongs to the class if and only if

Lemma 3.4. A function belongs to the class if and only if

Theorem 3.5. , where

Proof. If , we have which implies

Theorem 3.6. , where

Proof. If , we have which implies Thus, in view of condition (1.4), we get the required result of Theorem 3.6.

4. Neighborhood of the Class

Definition 4.1. A function is said to be in the class if it satisfies for , , and .

Theorem 4.2. For , one has and where

Proof. Let for . Then, Consider This implies that .

5. Distortion and Covering Theorems

Theorem 5.1. If , then with equality for

Proof. In view of Theorem 2.1, we have Hence, This completes the proof.

Theorem 5.2. Any function maps the disk onto a domain that contains the disk

Proof. The proof follows upon letting in Theorem 5.1.

Theorem 5.3. If , then with equality for

Proof. We have In view of Theorem 2.1, Thus, On the other hand, This completes the proof.

6. Radii of Starlikeness and Convexity

In this section, we find the radius of starlikeness of order and the radius of convexity of order for functions in the class .

Theorem 6.1. If , then is starlike of order , in , where

Proof. It is sufficient to show that for .
We have Thus, if Hence, by Theorem 2.1, (6.3) will be true if or if This completes the proof.

Theorem 6.2. If , then is convex of order , in , where

Proof. It is sufficient to show that for .
We have Thus, if Hence, by Theorem 2.1, (6.8) will be true if or if This completes the proof.

Acknowledgment

The author wish to thank the referee for his valuable suggestions.

References

  1. P. L. Duren, Univalent Functions, Springer, New York, NY, USA, 1983.
  2. F. M. Al-Oboudi, “On univalent functions defined by a generalized Sălăgean operator,” International Journal of Mathematics and Mathematical Sciences, no. 25–28, pp. 1429–1436, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. G. Sălăgean, “Subclasses of univalent functions,” in Complex analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Math., pp. 362–372, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. W. Goodman, “Univalent functions and nonanalytic curves,” Proceedings of the American Mathematical Society, vol. 8, pp. 598–601, 1957. View at Publisher · View at Google Scholar
  5. S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical Society, vol. 81, no. 4, pp. 521–527, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. K. Aouf, “Neighborhoods of certain classes of analytic functions with negative coefficients,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 38258, 6 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. I. S. Robertson, “On the theory of univalent functions,” Annals of Mathematics. Second Series, vol. 37, no. 2, pp. 374–408, 1936. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet