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Abstract and Applied Analysis
Volume 2010, Article ID 821531, 12 pages
http://dx.doi.org/10.1155/2010/821531
Research Article

A New Subclass of Salagean-Type Harmonic Univalent Functions

1Department of Applied Science, Ministry of Education, P.O. Box 75, P.C. 612, Bahla, Oman
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor Darul Ehsan, Bangi 43600, Malaysia
3Department of Mathematic, University of Ilorin, P.M.B. 1515, Ilorin, Nigeria

Received 5 November 2010; Accepted 31 December 2010

Academic Editor: Paul Eloe

Copyright © 2010 Khalifa Al-Shaqsi et al. 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 define and investigate a new subclass of Salagean-type harmonic univalent functions. We obtain coefficient conditions, extreme points, distortion bounds, convolution, and convex combination for the above subclass of harmonic functions.

1. Introduction

Let denote the class of functions of the form: which are analytic in the open unit disk .

We denote the subclass of consisting of analytic and univalent functions in the unit disk by .

The following classes of functions and many others are well known and have been studied repeatedly by many authors, namely, Sălăgean [1], Abdul Halim [2], and Darus [3] to mention but a few. (i). (ii). (iii). (iv).

In 1994, Opoola defined the class to be a subclass of consisting of analytic functions satisfying the condition where is the Salagean differential operator defined as follows: We note that is a generalization of the classes of functions , and .

Some properties of this class of functions were established by Opoola [4] namely, (i) is a subclass of univalent functions;(ii);(iii)if , then the integral operator is also in .

Now, by Binomial expansion, we have Hence, we define

2. Preliminaries

A continuous function is a complex-valued harmonic function in a 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 sense-preserving in is that in .

Denote by the class of functions of the form (2.1) that are harmonic univalent and sense-preserving in the unit disk . The subclasses of harmonic univalent functions have been studied by some authors for different purposes and different properties (see examples [512]). In this work, we may express the analytic functions and as Therefore, We define the modified Salagean operator of as where We let be the family of harmonic functions of the form (2.3) such that where is defined by (2.4).

It is clear that the class includes a variety of well-known subclasses of . For example, is the class of sense-preserving, harmonic univalent functions which are starlike of order in , that is, , and is the class of sense-preserving, harmonic univalent functions which are convex of order in , that is . Note that the classes and were introduced and studied by Jahangiri [5]. Also note that the class is the class of Salagean-type harmonic univalent functions introduced by Jahangiri et al. [13].

We let the subclass consist of harmonic functions in so and are of the form

In 1984, Clunie and Sheil-Small [14] investigated the class as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on and its subclasses such that Silverman [15], Silverman and Silvia [16], and Jahangiri [5, 17] studied the harmonic univalent functions. Jahangiri [5] proved the following theorem.

Theorem 2.1. Let where and . If then is sense-preserving, harmonic, and univalent in and . The condition (2.8) is also necessary if .

In this paper, we will give the sufficient condition for functions where and are given by (2.2) to be in the class and it is shown that these coefficient conditions are also necessary for functions in the class . Also, we obtain distortion theorems and characterize the extreme points for functions in . Convolution and convex combination are also obtained.

3. Main Results

In this section, we prove the main results.

3.1. Coefficient Estimates

Theorem 3.1. Let , where and are given by (2.2). If where , , , and , then is sense-preserving, harmonic univalent in , and .

Proof. If , then which proves univalence. Note that is sense-preserving in . This is because
By (2.6), Using the fact that if and only if , it suffices to show that Substituting for , in (3.6), we have This last expression is nonnegative by (3.1), and so the proof is complete.

The harmonic function where , , , and , shows that the coefficient bound given by (3.1) is sharp. The functions of the form (3.8) are in because

In the following theorem, it is shown that the condition (3.1) is also necessary for functions where and are of the form (2.7).

Theorem 3.2. Let be given by (2.7). Then , if and only if where , , , and .

Proof. Since , we only need to prove the “only if” part of the theorem. To this end, for functions of the form (2.7), we notice that the condition (2.6) is equivalent to The above required condition (3.11) must hold for all values of in . Upon choosing the values of on the positive real axis where , we must have If the condition (3.10) does not hold, then the numerator in (3.12) is negative for sufficiently close to 1. Hence there exist in for which the quotient in (3.12) is negative. This contradicts the required condition for and so the proof is complete.

3.2. Distortion Bounds and Extreme Points

In this section, first we will obtain distortion bounds for functions in .

Theorem 3.3. Let . Then for , we have

Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Let . Taking the absolute value of , we obtain for . This shows that the bounds given in Theorem 3.3 are sharp.

The following covering result follows from the left-hand inequality in Theorem 3.3.

Corollary 3.4. If function , where and are given by (2.7), is in , then

Next we determine the extreme points of closed convex hulls of denoted by .

Theorem 3.5. Let , where and are given by (2.7). Then if and only if where , , , and  ,  , . In particular, the extreme points of are and .

Proof. For functions , where and are given by (3.16), we have Then and so .
Conversely, suppose that . Setting and ; therefore, can be written as

3.3. Convolution and Convex Combination

In this section, we show that the class is invariant under convolution and convex combination of its member.

For harmonic functions and .

The convolution of and is given by

Theorem 3.6. For , let and . Then .

Proof. Let the functions be in the class and let the functions be in the class . Then the convolution is given by (3.21). We wish to show that the coefficients of satisfy the required condition given in Theorem 3.2. For , we note that and . Now, for the convolution function , we obtain since and . Therefore, .

We now examine the convex combination of .

Let the functions be defined, for , by

Theorem 3.7. Let the functions defined by (3.23) be in the class for every . Then the functions defined by are also in the class where .

Proof. According to the definition of , we can write Further, since are in for every , then by (3.1) we have Hence the theorem follows.

Corollary 3.8. The class is close under convex linear combination.

Proof. Let the functions defined by (3.23) be in the class . Then the function defined by is in the class . Also, by taking , , and in Theorem 3.7, we have the above corollary.

Acknowledgments

The first author was an ex-postgraduate student under the supervision of Professor Maslina Darus. Both the first and second authors are supported by UKM-ST-06-FRGS0107-2009. The third author made her contribution to the work while she was a Visiting Researcher at the African Institute for Mathematical Sciences, South Africa where she enjoyed numerous stimulating discussions with Professor Alan Beardon. The authors also would like to thank the anonymous referees for the informative and creative comments given to the paper.

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