Certain subclass of univalent functions involving fractional q-calculus operator

The main object of the present paper is to introduce certain subclass of univalent function associated with the concept of differential subordination. We studied some geometric properties likecoefficient inequality and nieghbourhood property, the Hadamard product properties and integraloperator meaninequality.


Introduction and Definitions
Let C be complex plane, letU denote the open unit disc in C, U = {z ∈ C : |z| < 1}, (1.1) and let Sbe the class of all analytic and univalent functions of the form For functions f and inS such that (z) defined by The Hadamard product(or convolution) of f and is defined by Now, we let f(z)and g(z)be members of H(U) .The function f(z) is said to be subordinate to a function g(z) or g(z) is said to be superordinate tof(z), if and only if there exists a Schwarz function w(z) analytic in U, with w(0) = 0 and |w(z)| < 1, (z ∈ U), such that Furthermore, if the function g is univalent in U, then we get the following equivalence f(z) ≺g(z) if and only if f(0) = g(0) and f(U) ⊂g(U) [3], [7].
The linear multiplier fractional q-differintegral operator ℒ , , introduced by [1] defined as follows. If f(z) is given by (1.4) then by (1.5), we get By (1.6) and (1.7), then we have Note that, if we put = 0 the operator ℒ , , reduces to the operator studied by AL-Oboudi [2] and for = 0, = 1, we get the operator introduced by S˘al˘agean [9].

Hadamard product properties
In this section we give some properties of the convolution belongs isthe concept. Proof. First, we find the largest so that ,2 ≤ 1.

Integral Mean Inequalities
In this section we study the integral meaninequalityby introduce the following definition.

Definition4.1.[10]:
The fractional integral of order s (s>0) is defined for a function f by: where the function f is an analytic in a simply connected region of the complex z-plane containing the origin, and multiplicity of − 1− is removed by requiring log(z-t) to be real, when (z-t) > 0.