Abstract and Applied Analysis

Volume 2013 (2013), Article ID 968627, 8 pages

http://dx.doi.org/10.1155/2013/968627

Research Article

Properties of a Class of -Harmonic Functions

Elif Yaşar and Sibel Yalçın

Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa, Turkey

Received 21 January 2013; Accepted 21 May 2013

Academic Editor: Youyu Wang

Copyright © 2013 Elif Yaşar and Sibel Yalçın. 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.

A times continuously differentiable complex-valued function in a domain is -harmonic if satisfies the -harmonic equation , where is a positive integer. By using the generalized Salagean differential operator, we introduce a class of -harmonic functions and investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points, and convex combination of the class.

1. Introduction

A continuous complex-valued function in a domain is harmonic if both and are real harmonic in ; that is, and . Here represents the complex Laplacian operator

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, 2].

Denote by the class of functions that are harmonic, univalent, and sense preserving in the unit disk for which . Then for we may express the analytic functions and as

The properties of the class and its geometric subclasses have been investigated by many authors; see ([1–6]). Note that reduces to the class of normalized analytic univalent functions in if the coanalytic part of is identically zero.

A times continuously differentiable complex-valued function in a domain is -harmonic if satisfies the -harmonic equation , where is a positive integer.

A function is -harmonic in a simply connected domain if and only if has the following representation:

where in for each . has the form

where

Denote by the class of functions of the form (3) that are harmonic, univalent, and sense-preserving in the unit disk. Apparently, if and ,   is harmonic and biharmonic, respectively.

Biharmonic functions have been studied by several authors, such as, [7–9]. Also, biharmonic functions arise in many physical situations, particularly, in fluid dynamics and elasticity problems. They have many important applications in engineering, biology, and medicine, such as in [10, 11].

For a function in , differential operator was introduced by Sălăgean [12]. Al-Oboudi [13] generalized as follows:

When , we get the Salagean differential operator.

For given by (2), Li and Liu [14] defined the following generalized Salagean operator in :

where

For a -harmonic function given by (3), we define the following operator:

If is given by (3), then from (10) we see that

When , we get the generalized Salagean operator for harmonic univalent functions defined by Li and Liu [14].

Denote by the class of functions of the form (3) which satisfy the condition

where is defined by (11).

We let the subclass of consist of functions of the form (3) which include , where

Define .

The main object of the paper is to introduce a class of -harmonic functions by using the generalized Salagean operator which was defined by Li and Liu [14]. We investigate necessary and sufficient coefficient conditions, extreme points, distortion bounds, and convex combination of the class.

2. Main Results

Theorem 1. Let be a -harmonic function given by (3). Furthermore, let

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

Proof. Suppose and , so that :

which proves univalence.

In order to prove that is sense preserving, we need to show that :

for all .

Using the fact that if and only if , it suffices to show that

Substituting for in (18), we obtain

This last expression is nonnegative by (15), and so the proof is complete.

Theorem 2. Let be given by (13) and (14). Then if and only if

where ,  ,  and .

Proof. The “if" part follows from Theorem 1 upon noting that . For the “only if" part, we show that if the condition (20) does not hold.

Note that a necessary and sufficient condition for given by (13) and (14) to be in is that the condition (12) should be satisfied.

This is equivalent to , where

The above condition must hold for all values of , . Upon choosing the values of on the positive real axis, where we must have

If the condition (20) does not hold, then the numerator in (22) is negative for is sufficiently close to . Hence there exist in for which the quotient in (22) is negative. This contradicts the required condition for and so the proof is complete.

Theorem 3. Let be given by (13) and (14). Then if and only if

where

and ,   ,  .

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

Proof. For functions of the form (13) and (14) we have

Then

and so . Conversely, if , then

Set

where . Then, as required, we obtain

Theorem 4. 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 have

The following covering result follows from the left-hand inequality in Theorem 4.

Corollary 5. Let of the form (13) and (14) be so that . Then

Theorem 6. The class is closed under convex combinations.

Proof. Let for , where is given by

Then by (20),

For , , the convex combination of may be written as

Then by (34),

This is the condition required by (20) and so .

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