International Journal of Analysis

Volume 2013 (2013), Article ID 352823, 7 pages

http://dx.doi.org/10.1155/2013/352823

Research Article

On Certain Classes of Harmonic -Valent Functions Defined by an Integral Operator

T. M. Seoudy

Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt

Received 3 November 2012; Accepted 18 December 2012

Academic Editor: Frédéric Robert

Copyright © 2013 T. M. Seoudy. 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 obtain coefficient characterization, extreme points, and distortion bounds of certain classes of harmonic -valent functions defined by an integral operator.

1. Introduction

A continuous complex-valued function defined in a simply connected complex domain is said to be harmonic in 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 (see [1]).

Denote by the class of functions of the form (1) that are harmonic univalent and sense preserving in the unit disc for which .

Recently, Jahangiri and Ahuja [2] defined the class , consisting of all -valent harmonic functions that are sense preserving in and , and are of the form

For given by (2), we define the modified -valent Salagean integral operator of (see [3] and also [4] when ) as follows:

where

For , and , we let denote the family of harmonic functions of the form (2) such that

where is defined by (3).

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

We note that , where the class was defined and studied by Cotirla [5].

In this paper, we obtain coefficient characterization of the classes and . We also obtain extreme points and distortion bounds for functions in the class .

2. Coefficient Characterization

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

Theorem 1. Let so that and are given by (2). Furthermore, let

where

Then, is sense preserving in and .

Proof. According to (2) and (3), we only need to show that

It follows that

For , we have

Setting that

the proof will be complete if we can show that . Using the condition (7), we can write

where

The harmonic functions are as follows:

where show that the coefficient bound given by (7) is sharp. The functions of the form (8) are in the class because

This completes the proof of Theorem 1.

In the following theorem, it is shown that the condition (7) is also necessary for functions , where and are of the form (6).

Theorem 2. Let , where and are given by (6). Then, if and only if

where and are given by (8) and (9), respectively.

Proof. Since , we only need to prove the “only if” part of the theorem. To this end, for functions , where and are given by (6), we notice that the condition is equivalent to

The previous required condition (19) must hold for all values of in . Upon choosing the values of on the positive real axis where , we must have

If the condition (18) does not hold, then the numerator in (20) is negative for sufficiently close to . Hence there exists in for which the quotient in (20) is negative. This contradicts the required condition for , and so the proof of Theorem 2 is completed.

3. Extreme Points and Distortion Theorem

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

Theorem 3. Let , where and are given by (6). Then, if and only if

where

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

Proof. Suppose that

Then,

and so .

Conversely, if , then

Set that

Then note that by Theorem 2, , and . We define that and note that by Theorem 2, . Consequently, we obtain as required.

The following theorem gives the distortion bounds for functions in the class which yields a covering result for this class.

Theorem 4. Let . Then, for , we have

where

The result is sharp.

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 have

The bounds given in Theorem 4 for functions , where and of form (6), also hold for functions of form (2) if the coefficient condition (7) is satisfied. The upper bound given for is sharp, and the equality occurs for the functions

showing that the bounds given in Theorem 4 are sharp.

Remark 5. (i) Putting in the previous results, we obtain the results of Cotirla [5].

(ii) Putting in the previous results, we obtain the results of Cotirla [6], when .

Acknowledgment

The author is grateful to the referees for their valuable suggestions.

References

J. Clunie and T. Sheil–Small, “Harmonic univalent functions,” Annales Academiæ Scientiarum Fennicæ Mathematica A, vol. 9, p. 3–25, 1984.

J. M. Jahangiri and O. P. Ahuja, “Multivalent harmonic starlike functions,” Annales Mariae Curie-Skłodowska A, vol. 56, pp. 1–13, 2001.

M. K. Aouf, A. O. Mostafa, and R. El-Ashwah, “Sandwich theorems for p-valent functions defined by a certain integral operator,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1647–1653, 2011.

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L.-I. Cotirla, “Harmonic multivalent functions defined by integral operator,” Studia Universitatis Babeş-Bolyai, vol. 54, no. 1, pp. 65–74, 2009.

L.-I. Cotirla, “A new class of harmonic multivalent functions defined by an integral operator,” Acta Universitatis Apulensis, vol. 21, pp. 55–63, 2010.