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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 509349, 12 pages
http://dx.doi.org/10.1155/2012/509349
Research Article

A Convolution Approach on Partial Sums of Certain Harmonic Univalent Functions

Department of Mathematics, UIET Campus, CSJM University, Kanpur 208024, India

Received 30 March 2012; Revised 13 October 2012; Accepted 14 October 2012

Academic Editor: S. S. Dragomir

Copyright © 2012 Saurabh Porwal. 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.

Abstract

The purpose of the present paper is to establish some new results giving the sharp bounds of the real parts of ratios of harmonic univalent functions to their sequences of partial sums by using convolution. Relevant connections of the results presented here with various known results are briefly indicated.

1. Introduction

A continuous complex-valued function is said to be harmonic in a simply connected 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 . A necessary and sufficient condition for to be locally univalent and sense-preserving in is that , see [1]. For more basic results on harmonic functions one may refer to the following standard text book by Duren [2]. See also Ahuja [3] and Ponnusamy and Rasila ([4, 5]).

Denote by the class of functions which are harmonic univalent and sense-preserving in the open unit disk for which . Then for we may express the analytic functions and as

Note that reduces to the class of normalized analytic univalent functions, if the coanalytic part of its member is zero, that is, , and for this class may be expressed as

Let be a fixed function of the form

Now, we introduce a class consisting of functions of the form (1.1) which satisfies the inequality and we note that if , then the class reduces to the class which was introduced by Frasin [6]. In this case the condition (1.4) reduces to

It is easy to see that various subclasses of consisting of functions of the form (1.1) can be represented as for suitable choices of , and studied earlier by various researchers. For example: (1) and studied by Silverman [7]; Silverman and Silvia [8]. (2) studied by Jahangiri [9]. (3) studied by Dixit and Porwal [10]. (4) studied by Dixit and Porwal [11]. (5) studied by Dixit and Porwal [12]. (6) studied by Dixit and Porwal [13]. (7) studied by Frasin [14]. (8) studied by Öztürk et al. [15]. (9) studied by Porwal et al. [16]. (10) studied by Rosy et al. [17]. In 1985, Silvia [18] studied the partial sums of convex functions of order . Later on, Silverman [19], Afaf et al. [20], Dixit and Porwal [21], Frasin ([6, 22]), Murugusundaramoorthy et al. [23], Owa et al. [24], Porwal and Dixit [25], Raina and Bansal [26] and Rosy et al. [27] studied and generalized the results on partial sums for various classes of analytic functions. Very recently, Porwal [28], Porwal and Dixit [29] studied analogues interesting results on the partial sums of certain harmonic univalent functions. In this work, we extend all these results.

Now, we let the sequences of partial sums of function of the form (1.1) with be when the coefficients of are sufficiently small to satisfy the condition (1.4).

In the present paper, we determine sharp lower bounds for , , , , , and where , and are defined above and , is a harmonic function and the operator “” stands for the Hadamard product or convolution of two power series, which is defined for two functions and are of the form as It is worthy to note that this study not only gives as a particular case, the results of Porwal [28], Porwal and Dixit [29], but also give rise to several new results.

2. Main Results

In our first theorem, we determine sharp lower bounds for .

Theorem 2.1. If of the form (1.1) with , satisfies the condition (1.4), then where
The result (2.1) is sharp with the function given by where .

Proof. To obtain sharp lower bound given by (2.1), let us put So that where denotes .
Then This last expression is bounded above by 1, if and only if It suffices to show that L.H.S. of (2.7) is bounded above by  , which is equivalent to
To see that gives the sharp result, we observe that for that when .

We next determine bounds for .

Theorem 2.2. If of the form (1.1) with , satisfies the condition (1.4), then where The result (2.10) is sharp with the function given by (2.3).

Proof. To prove Theorem 2.2, we may write where This last inequality is equivalent to Since the L.H.S. of (2.14) is bounded above by , the proof is evidently complete.

Adopting the same procedure as in Theorems 2.1 and 2.2 and performing simple calculations, we can obtain the sharp lower bounds for the real parts of the following ratios:

The results corresponding to real parts of these ratios are contained in the following Theorems 2.3, 2.4, 2.5, and 2.6.

Theorem 2.3. If of the form (1.1) with satisfies the condition (1.4), then where
The result (2.16) is sharp with the function

Theorem 2.4. If of the form (1.1) with , satisfies the condition (1.4), then where
The result (2.19) is sharp with the function given by (2.18).

Theorem 2.5. If of the form (1.1) with , satisfies the condition (1.4), then(i)where (ii)where The results (2.21) and (2.23) are sharp with the functions given by (2.3) and (2.18), respectively.

Theorem 2.6. If of the form (1.1) with , satisfies condition (1.4), then(i)where (ii)where The results (2.25) and (2.27) are sharp with the functions given by (2.3) and (2.18) respectively.

3. Some Consequences and Concluding Remarks

In this section, we specifically point out the relevances of some of our main results with those results which have appeared recently in literature.

If we put and in Theorems 2.12.6, then we obtain the corresponding results of Porwal [28].

Next, if we put , , , and in Theorems 2.12.6, then we obtain the corresponding results of Porwal and Dixit [29].

Again, if we put in Theorems 2.1 and 2.2, then we obtain the corresponding results of Dixit and Porwal [21].

Lastly, if we put , , and Theorems 2.1 and 2.2, then we obtain the result of Frasin [6].

We mention below some corollaries giving sharp bounds of the real parts on the ratio of univalent functions to its sequences of partial sums.

By putting in Theorem 2.1 for the function of the form (1.2) with and , then we obtain the following result of Silverman [19], Theorem .

Corollary 3.1. If of the form (1.2) satisfies the condition (1.5) with and , then The result is sharp for every , with the extremal function given by

On the other hand, if we put in Theorem 2.1 for the function of the form (1.2) with and , then we obtain the following result of Silverman, Theorem (i) [19].

Corollary 3.2. If of the form (1.2) satisfies the condition (1.5) with , then for The result is sharp for every , with the extremal function given by (3.2).

Also, if we put in Theorem 2.1 for the function of the form (1.2) belonging to the class , then we obtain the following result of Frasin [6].

Corollary 3.3. If , then where
The result is sharp for every , with the extremal function given by

Next, if we put in Theorem 2.1 for the function of the form (1.2) with and , then we obtain the following result of Murugusundaramoorthy et al. ([23], Theorem ).

Corollary 3.4. If of the form (1.2) satisfies the condition (1.5) with and , then for where
The result is sharp for every , with the extremal function given by

Again, if we set in Theorem 2.1, then we obtain the following result of Porwal [28].

Corollary 3.5. If of the form (1.1) with , satisfies the condition (1.4) with then The result (3.11) is sharp with the function given by (3.6).

Here we give some open problems for the readers.

In 2004, Owa et al. [24] studied the starlikeness and convexity properties on the partial sums and of the familiar Koebe function which is the extremal function for the class of starlike functions in the open unit disk and the function which is the extremal function for the class of convex functions in the open unit disk , respectively. They also presented some illustrative examples by using Mathematica (Version 4.0). It is interesting to obtain analogues results on harmonic starlikeness and convexity properties of the partial sums of the harmonic Koebe function.

In 2003, Jahangiri et al. [30] studied the construction of sense-preserving, univalent, and close-to-convex harmonic functions by using of the Alexander integral transforms of certain analytic functions (which are starlike or convex of positive order). They construct a function which is sense-preserving, univalent, and close-to-convex harmonic in , by using the result of Theorem [30] and taking the following function: It is worthy to note that the function (3.13) is of the form (3.6) with and . Therefore, it is natural to ask that the results of [30] may be generalized for the function of the form (3.6).

Acknowledgment

The author is thankful to the referee for his valuable comments and observations which helped in improving the paper.

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