Coefficient Estimates for Two New Subclasses of Biunivalent Functions with respect to Symmetric Points

We introduce two subclasses of biunivalent functions and find estimates on the coefficients and for functions in these new subclasses. Also, consequences of the results are pointed out.


Introduction and Definitions
Let  denote the class of analytic functions in the unit disk that have the form Further, by  we will denote the class of all functions in  which are univalent in .
If the functions  and  are analytic in , then  is said to be subordinate to , written as if there exists a Schwarz function (), analytic in , with such that Lewin [2] studied the class of biunivalent functions, obtaining the bound 1.51 for modulus of the second coefficient | 2 |.Subsequently, Netanyahu [3] showed that max | 2 | = 4/3 if () ∈ Σ. Brannan and Clunie [4] conjectured that | 2 | ≤ √ 2 for  ∈ Σ. Brannan and Taha [5] introduced certain subclasses of the biunivalent function class Σ similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike, and convex functions.They introduced bistarlike functions and obtained estimates on the initial coefficients.Bounds for the initial coefficients of several classes of functions were also investigated in [6][7][8][9][10][11][12][13][14][15].
Not much is known about the bounds on the general coefficient |  | for  ≥ 4. In the literature, there are only a few works determining the general coefficient bounds |  | for the analytic biunivalent functions ( [16][17][18][19][20] The classes  * () and () are the extensions of classical sets of starlike and convex functions and in such form were defined and studied by Ma and Minda [21].
and in the class   () if In this paper, we introduce two new subclasses of biunivalent functions.Further, we find estimates on the coefficients | 2 | and | 3 | for functions in these subclasses.
Definition 3. One notes that, for  = 0, one gets the class  *  (ℎ) which is defined as follows: where ℎ() and () satisfy the conditions of Definition 1. Furthermore, the functions ℎ() and () have the following Taylor-Maclaurin series expansions: respectively.From ( 15), we deduce From ( 18) and ( 20) we obtain By adding (19) to (21), we get Therefore, we find from ( 23) and ( 24 Subtracting ( 21) from ( 19) we have Then, upon substituting the value of  2 2 from ( 23) and ( 24) into (26), it follows that We thus find that This completes the proof of Theorem 4.
Taking  = 0 we get the following.
Definition 11.One notes that, for  = 0, one gets the class   (ℎ) which is defined as follows: (35)