ISRN Mathematical Analysis

Volume 2013 (2013), Article ID 387178, 5 pages

http://dx.doi.org/10.1155/2013/387178

Research Article

New Subclasses of Biunivalent Functions Involving Dziok-Srivastava Operator

M. K. Aouf,¹ R. M. El-Ashwah,² and Ahmed M. Abd-Eltawab³

¹Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

²Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

³Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt

Received 23 June 2013; Accepted 15 July 2013

Academic Editors: R. Avery, D. Bahuguna, and Y. Han

Copyright © 2013 M. K. Aouf et al. 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 introduce two new subclasses of biunivalent functions which are defined by using the Dziok-Srivastava operator. Furthermore, we find estimates on the coefficients and for functions in these new subclasses.

1. Introduction

Let denote the class of all functions of the form

which are analytic in the open unit disc . Also let denote the class of all functions in which are univalent in .

Some of the important and well-investigated subclasses of the univalent function class include, for example, the class of starlike functions of order in and the class of convex functions of order in . By definition, we have

Ding et al. [1] introduced the following class of analytic functions defined as follows:

It is easy to see that for . Thus, for , , and hence is univalent class (see [2–4]).

It is well known that every function has an inverse , defined by

where

A function is said to be bi-univalent in if both and are univalent in . Let denote the class of bi-univalent functions in given by (1). For a brief history and interesting examples in the class see [5].

Brannan and Taha [6] (see also [7]) introduced certain subclasses of the bi-univalent function class similar to the familiar subclasses and of starlike and convex functions of order , respectively (see [8]). Thus, following Brannan and Taha [6] (see also [7]), a function is in the class of strongly bi-starlike functions of order if each of the following conditions is satisfied:

where is the extension of to . The classes and of bi-starlike functions of order and biconvex functions of order , corresponding, respectively, to the function classes and , were also introduced analogously. For each of the function classes and , they found nonsharp estimates on the first two Taylor-Maclaurin coefficients and (for details, see [6, 7]).

For function given by (1) and given by

the Hadamard product (or convolution) of and is defined by

For complex parameters and , the generalized hypergeometric function is defined by the following infinite series:

where is the Pochhammer symbol (or shift factorial) defined, in terms of the Gamma function , by

Correspondingly a function is defined by

Dziok and Srivastava [9] (see also [10]) considered a linear operator

defined by the following Hadamard product:

If is given by (1), then we have

where

To make the notation simple, we write

It easily follows from (14) that

The linear operator is a generalization of many other linear operators considered earlier.

The object of the present paper is to introduce two new subclasses of the bi-univalent functions which are defined by using the Dziok-Srivastava operator and find estimates on the coefficients and for functions in these new subclasses of the function class employing the techniques used earlier by Srivastava et al. [5] (see also [11]).

In order to derive our main results, we have to recall here the following lemma [12].

Lemma 1. If , then for each , where is the family of all functions analytic in for which Re for .

Unless otherwise mentioned, we assume throughout this paper that , , ; , , , , is given by (15) and all powers are understood as principle values.

2. Coefficient Bounds of the Function Class

Definition 2. One says that a function given by (1) is said to be in the class if it satisfies the following condition:

where the function is given by

Remark 3. (i) For , , and , we have , where the class was introduced and studied by Frasin and Aouf [11].

(ii) For , , and , we have , where the class was introduced and studied by Srivastava et al. [5].

Theorem 4. Letting given by (1) be in the class , then

Proof. It follows from (18) that

where and in have the forms

Now, equating the coefficients in (22), we get

From (25) and (27), we get

Now from (26), (28), and (30), we obtain

Therefore, we have

Applying Lemma 1 for the coefficients and , we immediately have

This gives the bound on as asserted in (20).

Next, in order to find the bound on , by subtracting (28) from (26) and using (29), we get

It follows from (30) and (34) that

And, then,

Applying Lemma 1 once again for the coefficients , , , and , we readily get

This completes the proof of Theorem 4.

Remark 5. (i) Taking , , and , in Theorem 4, we obtain the result obtained by Frasin and Aouf [11, Theorem 2.2].

(ii) Taking , , and , in Theorem 4, we obtain the result obtained by Srivastava et al. [5, Theorem 1].

3. Coefficient Bounds of the Function Class

Definition 6. One says that a function given by (1) is said to be in the class if it satisfies the following condition:

where the function is defined by (19).

Remark 7. (i) For , , and , we have , where the class was introduced and studied by Frasin and Aouf [11].

(ii) For , , and , we have , where the class was introduced and studied by Srivastava et al. [5].

Theorem 8. Letting given by (1) be in the class , and , then

Proof. It follows from (38) that

where and have the forms (23) and (24), respectively.

As in the proof of Theorem 4, by suitably comparing coefficients in (41), we get

From (42) and (44), we get

Also, from (43) and (45), we find that

Therefore, we have

Applying Lemma 1 for the coefficients and , we immediately have

This gives the bound on as asserted in (39).

Next, in order to find the bound on , by subtracting (45) from (43), we get

or, equivalently,

and, then from (47), we find that

Applying Lemma 1 once again for the coefficients , , , and , we readily get

This completes the proof of Theorem 8.

Remark 9. (i) Taking , , and , in Theorem 8, we obtain the result obtained by Frasin and Aouf [11, Theorem 3.2].

(ii) Taking , , and , in Theorem 8, we obtain the result obtained by Srivastava et al. [5, Theorem 2].

References

S. S. Ding, Y. Ling, and G. J. Bao, “Some properties of a class of analytic functions,” Journal of Mathematical Analysis and Applications, vol. 195, no. 1, pp. 71–81, 1995.

M. P. Chen, “On the regular functions satisfying ,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 3, no. 1, pp. 65–70, 1975.

P. N. Chichra, “New subclasses of the class of close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 62, no. 1, pp. 37–43, 1976.

T. H. MacGregor, “Functions whose derivative has a positive real part,” Transactions of the American Mathematical Society, vol. 104, pp. 532–537, 1962.

H. M. Srivastava, A. K. Mishra, and P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1188–1192, 2010.

D. A. Brannan and T. S. Taha, “On some classes of bi-univalent functions,” in Mathematical Analysis and Its Applications, S. M. Mazhar, A. Hamoui, and N. S. Faour, Eds., vol. 3 of KFAS Proceedings Series, pp. 53–60, Pergamon Press, Oxford, UK, 1985, see also Studia Universitatis Babeş-Bolyai. Series Mathematica, vol. 31, no. 2, pp. 70–77, 1986.

T. S. Taha, Topics in univalent function theory [Ph.D. thesis], University of London, London, UK, 1981.

D. A. Brannan, J. Clunie, and W. E. Kirwan, “Coefficient estimates for a class of star-like functions,” Canadian Journal of Mathematics, vol. 22, pp. 476–485, 1970.

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.

J. Dziok and H. M. Srivastava, “Certain subclasses of analytic functions associated with the generalized hypergeometric function,” Integral Transforms and Special Functions, vol. 14, no. 1, pp. 7–18, 2003.

B. A. Frasin and M. K. Aouf, “New subclasses of bi-univalent functions,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1569–1573, 2011.

C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975.