Coefficients Bounds for Certain Subclass of Biunivalent Functions Associated with Ruscheweyh q-Differential Operator

1Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 2Department of Mathematics, Riphah International University, Islamabad, Pakistan 3School of Mathematical Sciences, Faculty of Sciences and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia 4Foundation Program, Dhofar University, Salalah, Oman 5Division of Engineering, Higher Colleges of Technology, P.O. Box 4114, Fujairah, UAE


Introduction
Let A denote the class of all function () which is analytic in the open unit disk  = { : || < 1} and has the Taylor series expansion of the following form: By S we mean the class of all functions in A which are univalent in .The Koebe one-quarter theorem [1] states that the image of  under every function  from S contains a disk of radius 1/4.It is well known that every univalent function  ∈ A has an inverse  −1 which is defined as where A function  ∈ A is said to be biunivalent in  if both  and  −1 are univalent in .
Using the technique of convolution, Ruscheweyh [22] defined the operator   on the class of analytic functions A as For  =  ∈  0 = {0}, we obtain The expression   () is called an th-order Ruscheweyh derivative of () and the symbol * stands for Hadamard product (or convolution).
For  ∈ R and  > 0,  ̸ = 1, the number [] is defined in [23] as For any nonnegative integer  the -number shift factorial is defined as We have lim →1 [] = .Throughout in this paper we will assume  to be fixed number between 0 and 1.
The -derivative operator or -difference operator for  ∈  is defined as It can easily be seen that for  ∈  = {1, 2, 3, . ..} and  ∈ .
The -generalized Pochhammer symbol for  ∈  and  ∈  is defined as and, for  > 0, let -gamma function be defined as For  ∈ A Ruscheweyh -differential operator was defined by Aldweby and Darus [24] (see also [23]), as If  → 1, equality (17) implies which is the well known recurrent formula for Ruscheweyh differential operator.in the present paper we introduce new subclass of the function class Σ, involving Ruscheweyh -differential operator    ().By using Faber polynomial coefficient techniques we determine estimates for the general coefficient bounds |  | for  ≥ 3 and also estimates on the coefficients | 2 | and | 3 | for functions in the new subclass of function class Σ.Several related classes are also considered, and connections to earlier known results are also defined. where and () =  −1 () is defined by (3).On specializing the parameters , , and , one can state the various new subclasses as illustrated in the following definition.
A function,  ∈ Σ, is in the class N
It is well known that Special Cases ; see [25].
By putting  = 1, in Theorem 6, we have the following corollary.

2𝑢 1 (
Journal of Complex AnalysisHere we give few examples of functions in the class Σ such that