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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 576571, 9 pages
http://dx.doi.org/10.1155/2008/576571
Review Article

On Some Subclasses of Harmonic Functions Defined by Fractional Calculus

Department of Mathematics, Faculty of Science, Girls College, P.O. Box 838, Dammam 31113, Saudi Arabia

Received 1 July 2008; Accepted 3 November 2008

Academic Editor: Teodor Bulboaca

Copyright © 2008 R. A. Al-Khal and H. A. Al-Kharsani. 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 purpose of this paper is to study subclasses of normalized harmonic functions with positive real part using fractional derivative. Sharp estimates for coefficients and distortion theorems are given.

1. Introduction

A continuous function is a complex-valued harmonic function in a complex domain if both and are real harmonic in . In any simply connected domain , we can write , where and are analytic in . We call the analytic part and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and orientation-preserving in is that in , see [1].

Denote by the class of functions which are harmonic univalent and orientation-preserving in the open unit disk so that is normalized by . Therefore, for , we can express and by the following power series expansion:Observe that reduces , the class of normalized univalent analytic functions, if the coanalytic part of is zero.

For given by (1.1) and , Murugusundaramoorthy [2] defined the Ruscheweyh derivative of the harmonic function in bywhere the Ruscheweh derivative of a power series is given by The operator stands for the Hadamard product or convolution of two power seriesdefined byIn [3], Owa introduced the following definition.

Definition 1.1. Let the function be analytic in a simply connected domain of the -plane containing the origin and let . The fractional derivative of of order is defined bywhere the multiplicity of is removed by requiring to be real when .

In [4], Owa gave the relation between the fractional derivative and Ruscheweyh operator for the function as

Using (1.2) and the relation between the fractional derivative and Ruscheweyh operator, we define the fractional derivative of order , , for the harmonic function asSince it was proved in [1] that the harmonic function is starlike of order if and only if the analytic function is starlike of order , and it was shown in [4, Theorem 3] that is starlike of order if and only if for . Since , then is starlike of order , hence is starlike of order . This means

Recently, Owa and Srivastava [5] studied the linear defined by operatorwhere is normalized and analytic function on .

It is easily seen thatAnalogously, we studied the linear operator defined on the harmonic function bywhereWe will define subclasses of normalized harmonic functions obtained by the Hadamard product and using the fractional derivative.

2. Main Results

Let and be analytic in . Let stand for harmonic functions so that and .

If the function belongs to for the analytic and normalized functionsthen the class of functions is denoted by [6].

The functionis analytic on when is a complex number different from For , we denote by the class of functions defined byTherefore,is in . Conversely, if is in the form (2.4), with being the coefficients of , then .

Note that [7] and .

Theorem 2.1. If , then there exists so thatConversely, for any function such that , there exists satisfying (2.5).

Proof. Let . If , then sinceas , we obtain thatTherefore,Conversely, for , from (2.1), (2.2), and (2.5),whereTherefore,

Theorem 2.2. A function of the form (2.4) belongs to , if and only if

Proof. If , then from Theorem 2.1,and . HenceConversely, if the function of the form (2.4) satisfies (2.10), then by Theorem 2.1   and the following function holds:Then by Theorem 2.1, .

Theorem 2.3. is convex and compact.

Proof. Let and let . ThenHence from Theorem 2.2, . Therefore, is convex.
Now, let and let . By Theorem 2.2,Since is compact, see [6],Hence by Theorem 2.1, , therefore is compact.

Theorem 2.4. If and , thenEquality is obtained for the function (2.3) where

Proof. From Theorem 2.1, if , then there exists so thatSince by [5, Proposition 2.2]this completes the proof.

Theorem 2.5. If and , then there exists so that

Proof. Sinceand for ,we haveHence is type (2.23).

Theorem 2.6. If , then .

Proof. Let and . Then there exists so thatHence and since , and . And hence .

Theorem 2.7. Let . Then
(i)
(ii) if is sense-preserving, then

Proof. By (2.10),Also, by [6, Theorem 2.3], we haveThe required results are obtained.
On the other hand, from (2.10), it is known [6, Corollary 2.5] thatThen we get the coefficient inequalities for .

Remark 2.8. Taking in Theorems 2.12.7, we get the similar results in [7].

Theorem 2.9. Let and sense-preserving in , then for ,

Proof. From Theorems 2.1 and 2.2, if , then there exists such thatBy [6, Theorem 3.5], we obtain the results.

Remark 2.10. Taking and in Theorem 2.9, we get [6, Theorem 2.4].

3. Positive Order

We say that the harmonic function of the form (2.1) is in the class for if and .

If the function belongs to for the analytic and normalized functions and of the form (2.1), then the class of functions is denoted by .

Denote by the class of functions defined By (2.3) where .

Many of our results can be rewritten for functions in the class . For instance, see the following theorems.

Theorem 3.1. If , then there exists so thatConversely, for any function such that , there exists satisfying (3.1).

Theorem 3.2. A function belongs to if and only if

Theorem 3.3. If and , then there exists so that

Theorem 3.4. If , then .

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