On q-Starlike Functions Defined by q-Ruscheweyh Differential Operator in Symmetric Conic Domain

Saira Zainab,Mohsan Raza,Sarfraz Nawaz Malik,Mehwish Jabeen,Sadia Riaz,Qin Xin

doi:10.3390/sym13101947

Abstract

Motivated by q-analogue theory and symmetric conic domain, we study here the q-version of the Ruscheweyh differential operator by applying it to the starlike functions which are related with the symmetric conic domain. The primary aim of this work is to first define and then study a new class of holomorphic functions using the q-Ruscheweyh differential operator. A new class k−STqτC,D of k-Janowski starlike functions associated with the symmetric conic domain, which are defined by the generalized Ruscheweyh derivative operator in the open unit disk, is introduced. The necessary and sufficient condition for a function to be in the class k−STqτC,D is established. In addition, the coefficient bound, partial sums and radii of starlikeness for the functions from the class of k-Janowski starlike functions related with symmetric conic domain are included.

Highlights

IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
We focus primarily on the partial sums and existence of the radius of the starlikeness of the k-Janowski starlike functions related to the generalized q-Ruscheweyh derivative
The following theorem gives a condition which is sufficient for functions to be in k − STqτ [C, D ]

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. After a careful study of the relevant literature, it was observed that the q-version of the well-known and most cited differential operator, named the Ruscheweyh differential operator, was introduced in [24] This has not been studied for starlike functions defined in the symmetric conic domain. The detailed study of the above-mentioned classes motivated us to define the much generalized class of functions with the q-Ruscheweyh differential operator related with the symmetric conic domain defined by Janowski functions. This class is denoted by k − STqτ [C, D ] and is defined as follows.

Main Results

Partial Sums

Conclusions