Certain Subclass of Univalent Functions Involving Fractional Q-Calculus Operator

Mustafa Ibrahim Hameed

doi:10.24297/jam.v13i4.6442

Abstract

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 like coefficient inequality and nieghbourhood property, the Hadamard product properties and integral operator mean inequality.

Highlights

Introduction and DefinitionsLet 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 f(z) = z + ∞ k =2 ak zk .(z ∈ U)For functions f and g inS such that g(z) defined by (1.2) g(z) = z + ∞ k =2 bk zkThe Hadamard product(or convolution) of f and g is defined by (z ∈ U)
Let K be a subset of S consisting of function f with the following form f(z) = z −
The linear multiplier fractional q-differintegral operator Lγα,τn introduced by [1] defined as follows

Summary

Introduction and Definitions

Let Sbe the class of all analytic and univalent functions of the form f(z) = z +. Let K be a subset of S consisting of function f with the following form f(z) = z −. If the function g is univalent in U, 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 Lγα,,τn introduced by [1] defined as follows. Definition 1.1.A function f z ∈ Kand θ ≥ 0 the θ– neighborhood f is defined as, NK,θ f = g z = z −. A function f(z)belonging to K is in the classK(α,λ,n,γ,τ,A,B),if it satisfies the following z Lγα,+τ λ,n f z ′′ 1 + Az. and is equivalent to the following condition:.

Main results

Hadamard product properties

Integral Mean Inequalities