COEFFICIENT ESTIMATES FOR A NEW SUBCLASS OF m-FOLD SYMMETRIC ANALYTIC BI-UNIVALENT FUNCTIONS

Considering a new subclass ofm-fold symmetric analytic bi-univalent functions, we determine estimates the coeffi cient bounds for the Taylor-Maclaurin coeffi cients |am+1| and |a2m+1| of the functions in this class. In certain cases, our estimates improve some of those existing coeffcient bounds.


Introduction
Let A denote the class of all functions of the form which are analytic in the open unit disk U = fz : z 2 C and jzj < 1g : We also denote by S the class of all functions in the normalized analytic function class A which are univalent in U.
For two functions f and , analytic in U; we say that the function f is subordinate to in U; and write if there exists a Schwarz function !; which is analytic in U with !(0) = 0 and j! (z)j < 1 (z 2 U) such that f (z) = (! (z)) (z 2 U) : It is well known that every function f 2 S has an inverse f 1 , which is de…ned by and f f 1 (w) = w jwj < r 0 (f ) ; r 0 (f ) 1 4 : In fact, the inverse function g = f 1 is given by g (w) = f 1 (w) = w a 2 w 2 + 2a 2 2 a 3 w 3 5a 3 2 5a 2 a 3 + a 4 w 4 + : (1.2) A function f 2 A is said to be bi-univalent in U if both f and f 1 are univalent in U; in the sense that f 1 has a univalent analytic continuation to U: We denote by the class of all bi-univalent functions in U given by (1:1).
For a brief history and interesting examples of functions in the class ; see [15] (see also [3]).In fact, the aforecited work of Srivastava et al. [15] essentially revived the investigation of various subclasses of the bi-univalent function class in recent years; it was followed by such works as those by Bulut et al. [5], Frasin and Aouf [6], Ramachandran et al. [9], Srivastava et al. [10,13], Xu et al. [18,19] and the references cited in each of them.
Let m 2 N = f1; 2; 3; : : :g : A domain D is said to be m-fold symmetric if a rotation of D about the origin through an angle 2 =m carries D on itself.It follows that, a function In particular, every f (z) is 1-fold symmetric and every odd f (z) is 2-fold symmetric.We denote by S m the class of m-fold symmetric univalent functions in U.
A simple argument shows that f 2 S m is characterized by having a power series of the form Srivastava et al. [14] de…ned m-fold symmetric bi-univalent functions analogues to the concept of m-fold symmetric univalent functions.For normalized form of f given by (1:3), they obtained the series expansion for f 1 as following: We denote by m the class of m-fold symmetric bi-univalent functions in U given by (1:3).For m = 1, the formula (1:4) coincides with the formula (1:2) of the class : For some examples of m-fold symmetric bi-univalent functions, see [14].
We also denote by P the family of all functions p analytic in U for which Thus the m-fold symmetric function p in the class P is of the form (see [8]), The coe¢ cient problem for m-fold symmetric analytic bi-univalent functions is one of the favorite subjects of geometric function theory in these days (see [1,4,7,11,14,16]).The object of the present paper is to introduce a new subclass of bi-univalent functions in which both f and f 1 are m-fold symmetric analytic functions and obtain coe¢ cient bounds for ja m+1 j and ja 2m+1 j for functions in this new subclass.
2. The Class N ;m ( ; ; ; ') Throughout this paper, we assume that ' is an analytic function with positive real part in the unit disk U; satisfying ' (0) = 1; ' 0 (0) > 0; and ' (U) is symmetric with respect to the real axis.Such a function has a series expansion of the form With this assumption on '; we now introduce the following new class of m-fold symmetric analytic bi-univalent functions.De…nition 1.For 2 Cn f0g ; 1 and 0 1; a function f 2 m given by (1:3) is said to be in the class N ;m ( ; ; ; ') if the following conditions are satis…ed: (ii ) There are many choices of the function ' (z) which would provide interesting subclasses of the analytic function class A. For example, if we let it is easy to verify that these functions are of the form (2:1) : If f 2 N ;m ( ; ; ; '), then f 2 m and w + g 0 (w) + wg 00 (w) 1 < 2 ; or where the function g = f 1 is de…ned by (1:4).This means that respectively.These classes are introduced and studied by Atshan and Al-Ziadi [2].
Corollary 1 is an improvement of the estimates obtained by Tang et al. [17, Theorem 7].