A Subclass of q-Starlike Functions Defined by Using a Symmetric q-Derivative Operator and Related with Generalized Symmetric Conic Domains

Shahid Khan,Muhammad Naeem,Saqib Hussain,Akhter Rasheed,Maslina Darus

doi:10.3390/math9090917

Abstract

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain Ωk,q,α˜, which generalizes the symmetric conic domains. By using the domain Ωk,q,α˜, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods.

Highlights

Let A be the set of all analytic functions in open unit disk U = {w ∈ C : |w| < 1} and every g ∈ A have the series representation of the form:
Later in [3], for k ≥ 0, Kanas and Wisniowska introduced the class of k-uniformly convex (k − U CV ) and k-uniformly starlike functions (k − U S T ) that are defined as: g ∈ k − U CV ⇐⇒ g ∈ A and Re 1 + wg (w) > k wg (w), w ∈ U
A function g ∈ A is said to be in class k − U S T (q, α), if it satisfies the condition:

Summary

Introduction

Let A be the set of all analytic functions in open unit disk U = {w ∈ C : |w| < 1} and every g ∈ A have the series representation of the form: Later in [3], for k ≥ 0, Kanas and Wisniowska introduced the class of k-uniformly convex (k − U CV ) and k-uniformly starlike functions (k − U S T ) that are defined as: g ∈ k − U CV ⇐⇒ g ∈ A and Re 1 + wg (w) > k wg (w) , w ∈ U (∂D)qwm = [m]qwm−1, (∂D)q{∞m=1amwm} =∞m=1 [m]qamwm−1, and: lim q→1−

Objectives

Results

Conclusion