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Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product


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



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

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Correspondence to H. Özlem Güney.

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Ahuja, O.P., Güney, H.Ö. & Sakar, F.M. Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product. J Inequal Appl 2009, 759251 (2009).

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  • Harmonic Function
  • Real Axis
  • Coefficient Estimate
  • Convex Combination
  • Multivalent Function