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# Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 759251 (2009)

## Abstract

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

then

(see [3]);

(see [4]);

(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

Proof.

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

Proof.

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.

Proof.

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

where

In particular, the extreme points of are and .

Proof.

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

Setting

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.

Proof.

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

Since

(2.27) yields

Thus the coefficient estimate given by Theorem 2.4 holds. Therefore, we obtain

## References

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## Acknowledgment

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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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). https://doi.org/10.1155/2009/759251

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### Keywords

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