Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product

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

(1.1)

are analytic functions in .

Let be a fixed multivalent harmonic function given by

(1.2)

A function is said to be in the class if

(1.3)

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

(1.4)

it follows that if and only if

(1.5)

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

(1.6)

then

(1.7)

(see [1, 2]);

(1.8)

(see [3]);

(1.9)

(see [4]);

(1.10)

(see [5, 6]);

(1.11)

(see [7]);

(1.12)

(see [8]).

Finally, denote by the subclass of functions in where

(1.13)

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

(2.1)

is satisfied for some and

Proof.

In view of (1.5), we need to prove that , where

(2.2)

Using the fact that , it sufficies to show that

(2.3)

Therefore, we obtain

(2.4)

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

The coeficient bounds (2.1) is sharp for the function

(2.5)

where .

Corollary 2.2.

For , , if the inequality

(2.6)

holds, then .

Corollary 2.3.

For and , if the inequality

(2.7)

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

(2.8)

(ii)for , if and only if

(2.9)

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

(2.10)

Choosing the values of on positive real axis where and using , the inequality (2.10) reduces to

(2.11)

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

Letting , we obtain

(2.12)

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

(2.13)

These bounds are sharp.

Proof.

Suppose . Let and . In view of (1.13), we get

(2.14)

Using Theorem 2.4(i), we obtain

(2.15)

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

Corollary 2.6.

If , then

(2.16)

Theorem 2.7.

Suppose and . Then clco if and only if

(2.17)

where

(2.18)

In particular, the extreme points of are and .

Proof.

Suppose . For functions of the form (2.17), we can write

(2.19)

On the other hand, for , we obtain

(2.20)

Thus , by Theorem 2.4.

Conversely, suppose that . Then, it follows Theorem 2.4 that

(2.21)

Setting

(2.22)

and defining

(2.23)

where , we obtain

(2.24)

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

(2.25)

are in . Also suppose the given fixed harmonic functions are given by

(2.26)

For the convex combinations of can be expressed as

(2.27)

Since

(2.28)

(2.27) yields

(2.29)

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

(2.30)