ISRN Mathematical Analysis

Volume 2013 (2013), Article ID 382312, 6 pages

http://dx.doi.org/10.1155/2013/382312

Research Article

A Subclass of Harmonic Univalent Functions Associated with -Analogue of Dziok-Srivastava Operator

Huda Aldweby and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 26 June 2013; Accepted 1 August 2013

Academic Editors: G. Ólafsson and D.-X. Zhou

Copyright © 2013 Huda Aldweby and Maslina Darus. 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.

We study a class of complex-valued harmonic univalent functions using a generalized operator involving basic hypergeometric function. Precisely, we give a necessary and sufficient coefficient condition for functions in this class. Distortion bounds, extreme points, and neighborhood of such functions are also considered.

1. Introduction

Let be the open unit disc, and let denote the class of functions which are complex valued, harmonic, univalent, and sense preserving in normalized by . Each can be expressed as , where and are analytic in . We call the analytic part and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and sense preserving in is that in (see [1]). In [2], there is a more comprehensive study on harmonic univalent functions. Thus, for , we may write

Note that reduces to , the class of normalized analytic univalent functions, if the coanalytic part of is identically zero.

The study of basic hypergeometric series (also called -hypergeometric series) essentially started in 1748 when Euler considered the infinite product . In the literature, we were told that the development of these functions was much slower until, in 1878, Heine converted a simple observation that which returns the theory of basic hypergeometric series to the famous theory of Gauss’s hypergeometric series. The importance of basic hypergeometric functions is due to their application in deriving -analogue of well-known functions, such as -analogues of the exponential, gamma, and beta functions. In this paper, we define a class of starlike harmonic functions using basic hypergeometric functions and investigate its properties like coefficient condition, distortion theorem, and extreme points.

For complex parameters , , , , , , we define the basic hypergeometric function by

where denote the set of positive integers and is the -shifted factorial defined by

We note that

where is the well-known generalized hypergeometric function. By the ratio test, one observes that for and the series defined in (2) converges absolutely in so that it represented an analytic function in . For more mathematical background of basic hypergeometric functions, one may refer to [3, 4].

The -derivative of a function is defined by

For a function , we can observe that

Then , where is the ordinary derivative. For more properties of , see [4, 5].

Corresponding to the function , consider

The authors [6] defined the linear operator by

where stands for convolution and

To make the notation simple, we write

We define the operator (8) of harmonic function given by (1) as

Definition 1. For , let denote the subfamily of starlike harmonic functions of the form (1) such that

Following [7], a function is said to be in the class if of the form (1) satisfies the condition that

and if there exists a real number such that

By specializing the parameters of , we obtain different classes of starlike harmonic functions, for example,

(i)for , [8] is the class of sense-preserving harmonic univalent functions which are starlike of order in ; that is, ;(ii)for , , and ,   [9] is the class of starlike harmonic univalent functions with , where is the Ruscheweyh derivative (see [10]); (iii)for , and , [11] is the class of starlike harmonic univalent functions with , where is the Dziok-Srivastava operator (see [12]).

2. Main Results

In our first theorem, we introduce a sufficient coefficient bound for harmonic functions in .

Theorem 2. Let be given by (1). If

where , and is given by (9), then .

Proof. To prove that , we only need to show that if (15) holds, then the required condition (12) is satisfied. For (12), we can write

Using the fact that if and only if , it suffices to show that

Substituting for and in (15) yields

The last expression is nonnegative by (15), and so, .

Now, we obtain the necessary and sufficient conditions for given by (14).

Theorem 3. Let be given by (11). Then, if and only if

where , and is given by (9).

Proof. Since , we only to prove the only if part of the theorem. So that for functions , we notice that the condition is equivalent to

That is,

The previous condition must hold for all values of in . Upon choosing according to (14), we must have

If condition (19) does not hold, then the numerator in (22) is negative for sufficiently close to 1. Hence, there exist in (0,1) for which the quotient of (22) is negative. This contradicts the fact that , and this completes the proof.

The following theorem gives the distortion bounds for functions in which yield a covering result for this class.

Theorem 4. If , then

where

Proof. We will only prove the right hand inequality. The proof for the left hand inequality is similar.

let . Taking the absolute value of , we obtain

That is,

Corollary 5. Let be of the form (1) so that . Then,

Next, one determines the extreme points of closed convex hull of denoted by .

Theorem 6. Set

For fixed, the extreme points for are

where and .

Proof. Any function may be expressed as

where the coefficients satisfy the inequality (15). Set , , , for . Writing , and ,  we have

In particular, set

Therefore, the extreme points of are contained in . To see that is not an extreme point, note that may be written as a convex linear combination of functions in as follows:

If both and , we will show that it can also be expressed as a convex linear combination of functions in . Without loss of generality, assume that . Choose small enough so that . Set and . We then see that both

are in and note that

The extremal coefficient bound shows that the functions of the form (29) are extreme points for , and so the proof is complete.

Following the earlier works in [2, 13], we refer to the -neighborhood of the functions defined by (1) to be the set of functions for which

We define the --neighborhood of a function as follows:

In our case, let us define the generalized --neighborhood of to be the set

Theorem 7. Let be given by (1). If satisfies the conditions

then .

Proof. Let satisfy (39) and let

belong to . We have

Hence,

Acknowledgment

The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02 and UKM-DLP-2011-050.

References

J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae A, vol. 9, pp. 3–25, 1984.

Y. Avcı and E. Złotkiewicz, “On harmonic univalent mappings,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 44, pp. 1–7, 1990.

G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990.

H. Exton, q-hypergeometric functions and applications, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood, Chichester, UK, 1983.

H. A. Ghany, “-derivative of basic hypergeometric series with respect to parameters,” International Journal of Mathematical Analysis, vol. 3, no. 33-36, pp. 1617–1632, 2009.

A. Mohammed and M. Darus, “A generalized operator involving the -hypergeometric function,” Matematicki Vesnik, vol. 65, no. 4, pp. 454–465, 2013.

J. M. Jahangiri and H. Silverman, “Harmonic univalent functions with varying arguments,” International Journal of Applied Mathematics, vol. 8, no. 3, pp. 267–275, 2002.

J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 470–477, 1999.

G. Murugusundaramoorthy, “A class of Ruscheweyh-type harmonic univalent functions with varying arguments,” Southwest Journal of Pure and Applied Mathematics, no. 2, pp. 90–95, 2003.

S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975.

H. A. Al-Kharsani and R. A. Al-Khal, “Univalent harmonic functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 2, article 59, p. 8, 2007.

J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,” Applied Mathematics and Computation, vol. 103, no. 1, pp. 1–13, 1999.

S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical Society, vol. 81, no. 4, pp. 521–527, 1981.