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

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)

Declarations

Acknowledgment

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

(1)
Department of Mathematical Sciences, Kent State University
(2)
Department of Mathematics, Faculty of Science and Arts, Dicle University

References

  1. Ahuja OP, Jahangiri JM: Multivalent harmonic starlike functions with missing coefficients. Mathematical Sciences Research Journal 2003,7(9):347–352.MathSciNetMATHGoogle Scholar
  2. Güney HÖ, Ahuja OP: Inequalities involving multipliers for multivalent harmonic functions. Journal of Inequalities in Pure and Applied Mathematics 2006,7(5, article 190):1–9.MathSciNetMATHGoogle Scholar
  3. Ahuja OP, Jahangiri JM: Multivalent harmonic starlike functions. Annales Universitatis Mariae Curie-Skłodowska. Sectio A 2001,55(1):1–13.MathSciNetMATHGoogle Scholar
  4. Jahangiri JM: Harmonic functions starlike in the unit disk. Journal of Mathematical Analysis and Applications 1999,235(2):470–477. 10.1006/jmaa.1999.6377MathSciNetView ArticleMATHGoogle Scholar
  5. Silverman H: Harmonic univalent functions with negative coefficients. Journal of Mathematical Analysis and Applications 1998,220(1):283–289. 10.1006/jmaa.1997.5882MathSciNetView ArticleMATHGoogle Scholar
  6. Silverman H, Silvia EM: Subclasses of harmonic univalent functions. New Zealand Journal of Mathematics 1999,28(2):275–284.MathSciNetMATHGoogle Scholar
  7. Ahuja OP, Aghalary R, Joshi SB: Harmonic univalent functions associated with k-uniformly starlike functions. Mathematical Sciences Research Journal 2005,9(1):9–17.MathSciNetMATHGoogle Scholar
  8. Jahangiri JM, Murugusundaramoorthy G, Vijaya K: On starlikeness of certain multivalent harmonic functions. Journal of Natural Geometry 2003,24(1–2):1–10.MathSciNetMATHGoogle Scholar

Copyright

© Om P. Ahuja et al. 2009

This article is published under license to BioMed Central Ltd. 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.