Open Access

Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product

Journal of Inequalities and Applications20092009:759251

DOI: 10.1155/2009/759251

Received: 25 February 2009

Accepted: 12 September 2009

Published: 29 September 2009


We define and investigate two special subclasses of the class of complex-valued harmonic multivalent functions based on Hadamard product.

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 co-analytic part of . Note that reduces to if the coanalytic part is zero.

For , denote by the set of all multivalent harmonic functions defined in the open unit disc , where and defined by


are analytic functions in .

Let be a fixed multivalent harmonic function given by


A function is said to be in the class if


where is a harmonic convolution of and . Note that . Using the fact


it follows that if and only if


A function in is called -uniformly multivalent harmonic starlike function associated with a fixed multivalent harmonic function . The set is a comprehensive family that contains several previously studied subclasses of ; for example, if we let



(see [1, 2]);

(see [3]);


(see [4]);


(see [5, 6]);


(see [7]);


(see [8]).

Finally, denote by the subclass of functions in where


Let .

In this paper, we investigate coefficient conditions, extreme points, and distortion bounds for functions in the family . We observe that the results so obtained for this main family can be viewed as extensions and generalizations for various subclasses of and .

2. Main Results

Theorem 2.1.

Let be such that and are given by (1.1). Then if the inequality

is satisfied for some and


In view of (1.5), we need to prove that , where
Using the fact that , it sufficies to show that
Therefore, we obtain

By hypothesis, last expression is nonnegative. Thus the proof is complete.

The coeficient bounds (2.1) is sharp for the function


where .

Corollary 2.2.

For , , if the inequality

holds, then .

Corollary 2.3.

For and , if the inequality

holds, then .

Theorem 2.4.

Let be such that and are given by (1.13). Also, suppose that and . Then

(i)for , if and only if


(ii)for , if and only if



According to Corollaries 2.2 and 2.3, we must show that if the condition (2.9) does not hold, then that is, we must have
Choosing the values of on positive real axis where and using , the inequality (2.10) reduces to

where denotes (k + )p[ + ], denotes (k+ )p + and denotes

Letting , we obtain

If the condition (2.10) does not hold, then the numerator in (2.12) is negative for sufficiently close to 1. Hence there exists in for which (2.12) is negative.Therefore, it follows that and so the proof is complete.

Theorem 2.5.

If , then for , and

These bounds are sharp.


Suppose . Let and . In view of (1.13), we get
Using Theorem 2.4(i), we obtain

The proofs of other cases are similar and so are omitted.

Corollary 2.6.

If , then

Theorem 2.7.

Suppose and . Then clco if and only if

In particular, the extreme points of are and .


Suppose . For functions of the form (2.17), we can write
On the other hand, for , we obtain

Thus , by Theorem 2.4.

Conversely, suppose that . Then, it follows Theorem 2.4 that
and defining
where , we obtain

Thus can be expressed as (2.17). The proof for the case is similar and hence is omitted.

Theorem 2.8.

The class is closed under convex combinations.


For let the functions given by
are in . Also suppose the given fixed harmonic functions are given by
For the convex combinations of can be expressed as
(2.27) yields
Thus the coefficient estimate given by Theorem 2.4 holds. Therefore, we obtain



This present investigation is supported with the Project no. DÜBAP-07-02-21 by Dicle University, The committee of the Scientific Research Projects.

Authors’ Affiliations

Department of Mathematical Sciences, Kent State University
Department of Mathematics, Faculty of Science and Arts, Dicle University


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© Om P. Ahuja et al. 2009

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