International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 915625, 10 pages

doi:10.1155/2012/915625

Research Article

Subclass of Multivalent Harmonic Functions with Missing Coefficients

R. M. El-Ashwah

Department of Mathematics, Faculty of Science (Damietta Branch), Mansoura University, New Damietta 34517, Egypt

Received 20 March 2012; Accepted 8 July 2012

Academic Editor: Attila Gilányi

Copyright © 2012 R. M. El-Ashwah. 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 have studied subclass of multivalent harmonic functions with missing coefficients in the open unit disc and obtained the basic properties such as coefficient characterization and distortion theorem, extreme points, and convolution.

1. Introduction

A continuous function is a complex-valued harmonic function in a simply connected complex domain if both and are real harmonic in . It was shown by Clunie and Sheil-Small [1] that such harmonic function can be represented by , where and are analytic in . Also, a necessary and sufficient condition for to be locally univalent and sense preserving in is that (see also, [2–4]).

Denote by the family of functions , which are harmonic univalent and sense-preserving in the open-unit disc with normalization .

For andlet denote the class of all multivalent harmonic functions with missing coefficients that are sense-preserving in , and are of the form

and satisfying the following condition:

where

We note that:

(i) with (see Jahangiri et al. [5]);(ii) (see Kharinar and More [6]);(iii)with (see Rosy et al. [7] and Ahuja and Jahangiri [2]).

Also, the subclass denoted by consists of harmonic functions , so that and are of the form

We note that:

(i) with (see Jahangiri et al. [5]);(ii)(see Kharinar and More [6]);(iii) with (see Rosy et al. [7] and Ahuja and Jahangiri [2]).

From Ahuja and Jahangiri [2] with slight modification and among other things proved, if is of the form (1.1) and satisfies the coefficient condition

then the harmonic function is sense-preserving, harmonic multivalent with missing coefficients and starlike of order in They also proved that the condition (1.5) is also necessary for the starlikeness of function of the form (1.4).

In this paper, we obtain sufficient coefficient bounds for functions in the class . These sufficient coefficient conditions are shown to be also necessary for functions in the class . Basic properties such as distortion theorem, extreme points, and convolution for the class are also obtained.

2. Coefficient Characterization and Distortion Theorem

Unless otherwise mentioned, we assume throughout this paper that andis real. We begin with a sufficient condition for functions in the class .

Theorem 2.1. Let be such that and are given by (1.1). Furthermore, let

where Then is sense-preserving, harmonic multivalent in and .

Proof. To prove , by definition, we only need to show that the condition (2.1) holds for . Substituting for in (1.2), it suffices to show that

where and . Substituting for , and in (2.2), and dividing by , we obtain , where

Using the fact that if and only if in , it suffices to show thatSubstituting for and gives

The harmonic functions

where , show that the coefficient boundary given by (2.1) is sharp. The functions of the form (2.5) are in the class because

This completes the proof of Theorem 2.1.

In the following theorem, it is shown that the condition (2.1) is also necessary for functions , where and are of the form (1.4).

Theorem 2.2. Let be such that and are given by (1.4). Then if and only if

Proof. Since , we only need to prove the “only if” part of the theorem. To this end, for functions of the form (1.4), we notice that the condition is equivalent to

which implies that

Since , the required condition is that (2.9) is equivalent to

If the condition (2.7) does not hold, then the numerator in (2.10) is negative for sufficiently close to . Hence there exists in for which the quotient in (2.10) is negative. This contradicts the required condition for , and so the proof of Theorem 2.2 is completed.

Corollary 2.3. The functions in the class are starlike of order .

Proof. The proof follows from (1.5), by putting (2.7) in the form

Theorem 2.4. Let . Then for , we have

Proof. We prove the left-hand-side inequality for The proof for the right-hand-side inequality can be done by using similar arguments.

Let , then we have

This completes the proof of Theorem 2.4.

The following covering result follows from the left-side inequality in Theorem 2.4.

Corollary 2.5. Let then the set

is included in .

3. Extreme Points

Our next theorem is on the extreme points of convex hulls of the class , denoted by .

Theorem 3.1. Let be such that and are given by (1.4). Then if and only if can be expressed as

where

In particular, the extreme points of the class are and respectively.

Proof. For functions of the form (3.1), we have

Then

and so Conversely, suppose that . Set

then note that by Theorem 2.2, and .

Consequently, we obtain

Using Theorem 2.2 it is easily seen that the class is convex and closed, and so .

4. Convolution Result

For harmonic functions of the form

we define the convolution of two harmonic functions and as

Using this definition, we show that the class is closed under convolution.

Theorem 4.1. For , let and . Then .

Proof. Let the functions defined by (4.1) be in the class, and let the functions defined by (4.2) be in the class . Obviously, the coefficients of and must satisfy a condition similar to the inequality (2.7). So for the coefficients of we can write

where the right hand side of this inequality is bounded by 2 because . Then, .

Finally, we show that is closed under convex combinations of its members.

Theorem 4.2. The class is closed under convex combination.

Proof. For . let where the functions are given by

For the convex combination of may be written as

Then by (2.7), we have

This is the condition required by (2.7), and so This completes the proof of Theorem 4.2.

Remark 4.3. Our results for correct the results obtained by Kharinar and More [6].

Acknowledgment

The author thanks the referees for their valuable suggestions which led to the improvement of this study.

References

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