Coefficient Bounds for Certain Subclasses of m-Fold Symmetric Biunivalent Functions

We consider two new subclasses and of consisting of analytic and -fold symmetric biunivalent functions in the open unit disk . Furthermore, we establish bounds for the coefficients for these subclasses and several related classes are also considered and connections to earlier known results are made.


Introduction
Let  denote the class of functions of the form which are analytic in the open unit disk  = { : || < 1}, and let  be the subclass of  consisting of form (1) which is also univalent in .
For a brief history and interesting examples in class Σ, see [2].Examples of functions in class and so on.However, the familiar Koebe function is not a member of Σ.Other common examples of functions in  such as are also not members of Σ (see [2]).
For each function  ∈ , function is univalent and maps unit disk  into a region with -fold symmetry.A function is said to be -fold symmetric (see [3,4]) if it has the following normalized form: 2

Journal of Mathematics
We denote by   the class of -fold symmetric univalent functions in , which are normalized by the series expansion (7).In fact, the functions in class  are one-fold symmetric.
Analogous to the concept of -fold symmetric univalent functions, we here introduced the concept of -fold symmetric biunivalent functions.Each function  ∈ Σ generates an -fold symmetric biunivalent function for each integer  ∈ N. The normalized form of  is given as in (7) and the series expansion for  −1 , which has been recently proven by Srivastava et al. [5], is given as follows: where  −1 = .We denote by Σ  the class of -fold symmetric biunivalent functions in .For  = 1, formula (8) coincides with formula (3) of class Σ.Some examples of -fold symmetric biunivalent functions are given as follows: Lewin [6] studied the class of biunivalent functions, obtaining the bound 1.51 for modulus of the second coefficient | 2 |.Subsequently, Brannan and Clunie [7] conjectured that | 2 | ≤ √ 2 for  ∈ Σ.Later, Netanyahu [8] showed that max | 2 | = 4/3 if () ∈ Σ. Brannan and Taha [9] introduced certain subclasses of biunivalent function class Σ similar to the familiar subclasses. ⋆ () and () are of starlike and convex function of order  (0 ≤  < 1), respectively (see [8]).Classes  ⋆ Σ () and  Σ () of bistarlike functions of order  and biconvex functions of order , corresponding to function classes  ⋆ () and (), were also introduced analogously.For each of function classes  ⋆ Σ () and  Σ (), they found nonsharp estimates on the initial coefficients.In fact, the aforecited work of Srivastava et al. [2] essentially revived the investigation of various subclasses of biunivalent function class Σ in recent years.Recently, many authors investigated bounds for various subclasses of biunivalent functions (see [2,[10][11][12][13][14][15]).Not much is known about the bounds on general coefficient |  | for  ≥ 4. In the literature, only few works determine general coefficient bounds |  | for the analytic biunivalent functions (see [16][17][18]).The coefficient estimate problem for each of The aim of the this paper is to introduce two new subclasses of function class Σ  and derive estimates on initial coefficients | +1 | and | 2+1 | for functions in these new subclasses.We have to remember the following lemma here so as to derive our basic results.Lemma 1 (see [4]).