Certain Subclasses of Bistarlike and Biconvex Functions Based on Quasi-Subordination NanjundanMagesh

and Applied Analysis 3 where Fλ (z) = (1 − λ) f (z) + λzf 󸀠 (z) , Gλ (w) = (1 − λ) g (w) + λwg 󸀠 (w)


Introduction
Let A denote the class of functions of the form where all coefficients are real.Also, let  be an analytic and univalent function with positive real part in U with (0) = 1,   (0) > 0 and  maps the unit disk U onto a region starlike with respect to 1 and symmetric with respect to the real axis.
Taylor's series expansion of such function is of the form where all coefficients are real and  1 > 0. Throughout this paper we assume that the functions ℎ and  satisfy the above conditions one or otherwise stated.
For two functions  and  are analytic in U, we say that the function () is subordinate to () in U and write if there exists a Schwarz function (), analytic in U, with such that In particular, if the function  is univalent in U, the above subordination is equivalent to  (0) =  (0) ,  (U) ⊂  (U) .
In [2] Ma and Minda introduced the unified classes S * () and K() given below: For the choice or the classes S * () and K() consist of functions known as the starlike (resp., convex) functions of order  or strongly starlike (resp., convex) functions of order , respectively.Recently, El-Ashwah and Kanas [3] introduced and studied the following two subclasses: We note that when ℎ() ≡ 1, the classes S *  (, ) and K  (, ) reduce, respectively, to the familiar classes S * (, ) and K(, ) of Ma-Minda starlike and convex functions of complex order  ( ∈ C \ {0}) in U (see [4]).For  = 1, the classes S *  (, ) and K  (, ) reduce to the classes S *  () and K  (), respectively, that are analogous to Ma-Minda starlike and convex functions, introduced by Mohd and Darus [5].
It is well known that every function  ∈ S has an inverse  −1 , defined by where where and |  | is the ( − 1)th order determinant whose entries are defined in terms of the coefficients of () by the following: For initial values of , we have A function  ∈  given by ( 1) is said to be in the class M , , (, ), 0 ̸ =  ∈ C,  ≥ 0, if the following quasi-subordination conditions are satisfied: Abstract and Applied Analysis 3 where and the function  is the extension of  −1 to U.
It is interesting to note that the special values of , , , and  and the class M , , (, ) unify the following known and new classes.
In this paper we introduce the unified biunivalent function class M , , (, ) as defined above and obtain the coefficient estimates for Taylor-Maclaurin coefficients | 2 | and | 3 | for functions belonging to M , , (, ).Some interesting applications of the results presented here are also discussed.
In order to derive our results, we need the following lemma.
In light of Remarks 1-5, we have following corollaries.
) which are analytic in the open unit disc U = { :  ∈ C and || < 1}.Further, by S we denote the family of all functions in A which are univalent in U. Let ℎ() be an analytic function in U and |ℎ()| ≤ 1, such that ) in the class M   (), we are led to the class which we denote, for convenience, by M   (), introduced and studied by Li and Wang [12, Definition 3.1.,p. 1500], and upon replacing  by (13) in the class M   (), we have M   (); this class was introduced and studied by Li and Wang [12, Definition 2.1.,p. 1497].Remark 2. Taking  = 0 and  = 0 in the class M , , (, ), we have M 0,0 , (, ) fl S * , (, ) .