International Journal of Mathematics and Mathematical Sciences

Volume 2011 (2011), Article ID 103521, 10 pages

http://dx.doi.org/10.1155/2011/103521

## 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.

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