Abstract and Applied Analysis

Volume 2013 (2013), Article ID 573017, 3 pages

http://dx.doi.org/10.1155/2013/573017

## Coefficient Bounds for Certain Subclasses of Bi-Univalent Function

^{1}School of Advanced Sciences, VIT University, Vellore, Tamil Nadu 632014, India^{2}PG and Research Department of Mathematics, Government Arts College (Men), Krishnagiri, Tamil Nadu 635001, India^{3}Department of Mathematics, Adhiyamaan College of Engineering (Autonomous), Hosur, Tamil Nadu 635109, India

Received 11 February 2013; Accepted 23 May 2013

Academic Editor: Pavel Kurasov

Copyright © 2013 G. Murugusundaramoorthy 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 the function class Σ of bi-univalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients and for functions in these new subclasses. Also consequences of the results are pointed out.

#### 1. Introduction

Denote by the class of functions of the form which are analytic in the open unit disc . Further, denote by the class of all functions in which are univalent and normalized by in . The well-investigated subclasses of the univalent function class are the class of starlike functions of order , denoted by and the class of convex functions of order denoted by in . 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).

Analogous to the function class , the bi-univalent function class include, for example, the class of bi-starlike functions of order , the class of biconvex functions of order , and the class of strongly bi-starlike functions of order . For some intriguing examples of functions and characterization of the class , one could refer to Srivastava et al. [1] and the references stated therein (see also [2]). Recently there has been triggering, interest to study the bi-univalent functions class (see [2–5]) and obtain nonsharp estimates on the first two Taylor-Maclaurin coefficients and . The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients for , is presumably still an open problem.

Motivated by the earlier works of Srivastava et al. [1] and Frasin and Aouf [3] in the present paper we introduce the following two new subclasses of the function class .

*Definition 1. *A function given by (1) is said to be in the class if the following conditions are satisfied:
where the function is given by

*Definition 2. *A function given by (1) is said to be in the class if the following conditions are satisfied:
where the function is given by (5).

It is of interest to note that, for , the class reduces to of strongly bi-starlike functions of order and the class leads to bi-starlike functions of order .

The object of the present paper is to find estimates on the coefficients and for functions in the above-defined subclasses and of the function class by employing the techniques used earlier by Srivastava et al. [1].

In order to derive our main results, we recall the following lemma.

Lemma 3 (see [6]). *If , then for each , where , is the family of all functions analytic in for which , where for .*

#### 2. Coefficient Bounds for the Function Class

We begin by finding the estimates on the coefficients and for functions in the class .

Theorem 4. *Let given by (1) be in the class , , and . Then
*

*Proof. *It follows from (4) that
where and in have the forms
Now, equating the coefficients in (9), we get
From (12) and (14), we get
From (13), (15), and (17), we obtain
Applying Lemma 3 for the coefficients and , we immediately have
This gives the bound on as asserted in (7).

Next, in order to find the bound on , by subtracting (15) from (13), we get
It follows from (16), (17), and (20) that
Applying Lemma 3 once again for the coefficients and , we readily get
This completes the proof of Theorem 4.

In the following section we find the estimates on the coefficients and for functions in the class .

#### 3. Coefficient Bounds for the Function Class

Theorem 5. *Let given by (1) be in the class , and . Then
*

*Proof. *It follows from (6) that there exists such that
where and have the forms of (10) and (11), respectively. Equating coefficients in (24) we get

The proof follows, by employing the techniques used in the proof of Theorem 4.

Taking in Theorems 4 and 5 one can get the following corollaries.

Corollary 6. *Let given by (1) be in the class and . Then
*

Corollary 7. *Let given by (1) be in the class and . Then
*

#### Acknowledgment

The authors would like to record their sincere thanks to the referees for their valuable suggestions.

#### References

- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Hayami and S. Owa, “Coefficient bounds for bi-univalent functions,”
*Pan-American Mathematical Journal*, vol. 22, no. 4, pp. 15–26, 2012. View at Google Scholar - B. A. Frasin and M. K. Aouf, “New subclasses of bi-univalent functions,”
*Applied Mathematics Letters*, vol. 24, no. 9, pp. 1569–1573, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-H. Xu, Y.-C. Gui, and H. M. Srivastava, “Coefficient estimates for a certain subclass of analytic and bi-univalent functions,”
*Applied Mathematics Letters*, vol. 25, no. 6, pp. 990–994, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-H. Xu, H.-G. Xiao, and H. M. Srivastava, “A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems,”
*Applied Mathematics and Computation*, vol. 218, no. 23, pp. 11461–11465, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - C. Pommerenke,
*Univalent Functions*, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975. View at MathSciNet