Abstract

Motivated and stimulated especially by the work of Xu et al. [1] , in this paper, we introduce and discuss an interesting subclass of analytic and bi-univalent functions defined in the open unit disc. Further, we find estimates on the coefficients and for functions in this subclass. Many relevant connections with known or new results are pointed out.

Keywords:Analytic Functions, Univalent Functions, Bi-Univalent Functions, Bi-Starlike Functions

1. Introduction

Let denote the class of functions of the form

(1.1)

which are analytic in the open unit disc Further, by we shall 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 strongly starlike functions of order in

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

and

where

(1.2)

A function is said to be bi-univalent in if both and are univalent in We denote by the class of all bi-univalent functions in For a brief history and interesting examples of functions in the class see [2] and the references therein.

In fact, the study of the coefficient problems involving bi-univalent functions was revived recently by Srivastava et al. [2] . Various subclasses of the bi-univalent function class were introduced and non-sharp estimates on the first two Taylor-Maclaurin coefficients and of functions in these subclasses were found in several recent investigations (see, for example, [3] -[13] ). The aforecited all these papers on the subject were motivated by the pioneering work of Srivastava et al. [2] . But the coefficient problem for each of the following Taylor-Maclaurin coefficients () is still an open problem.

Motivated by the aforecited works (especially [1] ), we introduce the following subclass of the analytic function class

Definition 1 Let and the functions be so constrained that

and We say that if the following conditions are satisfied:

(1.3)

and

(1.4)

where the function is the extension of to

We note that, for the different choices of the functions and, we get interesting known and new subclasses of the analytic function class For example, if we set

in the class then we have Also, if the following conditions are satisfied:

and

where is the extension of to

Similarly, if we let

in the class then we get Further, we say that if the following conditions are satisfied:

and

where is the extension of to

The classes and were introduced and studied by Murugusundaramoorthy et al. [12] , Definition 1.1 and Definition 1.2]. The classes and are strongly bi-starlike functions of order and bi-starlike functions of order respectively. The classes and were introduced and studied by Brannan and Taha [14] , Definition 1.1 and Definition 1.2]. In addition, we note that, was introduced and studied by Bulut [4] , Definition 3].

Motivated and stimulated by Bulut [4] and Xu et al. [1] (also [10] ), in this paper, we introduce a new subclass and obtain the estimates on the coefficients and for functions in aforementioned class, employing the techniques used earlier by Xu et al. [1] .

2. A Set of General Coefficient Estimates

In this section we state and prove our general results involving the bi-univalent function class given by Definition 1.

Theorem 1 Let be of the form (1.1). If then

(1.5)

and

(1.6)

Proof 1 Since From (1.3) and (1.4), we have,

and

where

and

satisfy the conditions of Definition 1. Now, upon equating the coefficients of with those of and the coefficients of with those of, we get

(1.7)

(1.8)

(1.9)

and

(1.10)

From (1.7) and (1.9), we get

(1.11)

and

(1.12)

From (1.8) and (1.10), we obtain

(1.13)

Therefore, we find from (1.12) and (1.13) that

(1.14)

and

(1.15)

Since and we immediately have

and

respectively. So we get the desired estimate on as asserted in (1.5).

Next, in order to find the bound on, by subtracting (1.10) from (1.8), we get

(1.16)

Upon substituting the values of from (1.14) and (1.15) into (1.16), we have

and

respectively. Since and we readily get

and

This completes the proof of Theorem 1.

If we choose

in Theorem 1, we have the following corollary.

Corollary 1 Let be of the form (1.1) and in the class Then

and

If we set

in Theorem 1, we readily have the following corollary.

Corollary 2 Let be of the form (1.1) and in the class Then

and

Remark 1 The estimates on the coefficients and of Corollaries 1 and 2 are improvement of the estimates obtained in [10] , Theorems 4 and 5]. Taking in Corollaries 1 and 2, the estimates on the coefficients and are improvement of the estimates in [14] , Theorems 2.1 and 4.1]. When the results discussed in this article reduce to results in [4] . Similarly, various other interesting corollaries and consequences of our main result can be derived by choosing different and.

Acknowledgements

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

Funding

The work is supported by UGC, under the grant F.MRP-3977/11 (MRP/UGC-SERO) of the first author.

References