Coefficient Estimates for a Certain General Subclass of Analytic and Bi-univalent Functions

Motivated and stimulated especially by the work of Xu et al. [1], in this paper, we introduce and discuss an interesting subclass () ,  Σ ϕ ψ λ of analytic and bi-univalent functions defined in the open unit disc . Further, we find estimates on the coefficients 2 a and 3 a for functions in this subclass. Many relevant connections with known or new results are pointed out.


Introduction
Let  denote the class of functions of the form ( )  A function f ∈  is said to be bi-univalent in  if both f and 1 f − are univalent in . We denote by Σ the class of all bi-univalent functions in .
 For a brief history and interesting examples of functions in the class Σ 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 Σ were introduced and non-sharp estimates on the first two Taylor-Maclaurin coefficients 2 a and 3 a 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 n a ( ) 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 f ∈  and the functions , : and where the function g is the extension of 1 f − to . We note that, for the different choices of the functions ϕ and ψ , we get interesting known and new sub- classes of the analytic function class . For example, if we set ( ) ( ) ( ) if the following conditions are satisfied: where g is the extension of where g is the extension of was introduced and studied 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 2 a and 3 a for functions in aforementioned class, employing the techniques used earlier by Xu et al. [1].

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.

( )
f z be of the form (1.1) and in the class in Theorem 1, we readily have the following corollary.The estimates on the coefficients 2 a and 3 a of Corollaries 1 and 2 are improvement of the estimates obtained in [10], Theorems 4 and 5].Taking 0 λ = in Corollaries 1 and 2, the estimates on the co- efficients 2 a and 3 a are improvement of the estimates in [14], Theorems 2.1 and 4.1].When 0 λ = 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 ϕ and ψ .
 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 ( ) * β  of starlike functions of order β ( ) in  and the class ( ) * α  of strongly starlike functions of order α ( ) It is well known that every function f ∈  has an inverse 1 , f − defined by

1 f
were introduced and studied by Murugusundaramoorthy et al.[12], Definition 1.1 and Definition 1.2].The classes functions of order α and bi-starlike functions of order β respectively.The classes were introduced and studied by Brannan and Taha[14], Definition 1.1 and Definition 1.2].In addition, we note that, This completes the proof of Theorem 1.