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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 103521, 10 pages
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.
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.
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 .
Definition 1.1 (see ). For and , the Al-Oboudi operator is defined as , , and .
For , we get Sǎlǎgean differential operator .
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  and then generalized by Ruscheweyh .
Definition 1.2. A function is said to be in the class if where , , , and .
2. Neighborhoods for the Class
Theorem 2.1. A function belongs to the class if and only if for , , , and .
Proof. Let . Then,
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
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 .
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
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
Theorem 6.2. If , then is convex of order , in , where
The author wish to thank the referee for his valuable suggestions.
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