Some General Classes of q-Starlike Functions Associated with the Janowski Functions

Hari Srivastava,Nazar Khan,Qazi Ahmad,Muhammad Tahir,Bilal Khan

doi:10.3390/sym11020292

Abstract

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems.

Highlights

The basic calculus is the ordinary classical calculus without the notion of limits, while q stands for the quantum
Mohammed and Darus [10] studied the approximation and geometric properties of these q-operators in some subclasses of analytic functions in a compact disk. These q-operators are defined by using the convolution of normalized analytic functions and q-hypergeometric functions, where several interesting results were obtained
Thereafter we will demonstrate three subclasses of the class Sq∗ of q-starlike functions associated with the Janowski functions

Summary

Introduction

The basic (or q-) calculus is the ordinary classical calculus without the notion of limits, while q stands for the quantum. Mohammed and Darus [10] studied the approximation and geometric properties of these q-operators in some subclasses of analytic functions in a compact disk. These q-operators are defined by using the convolution of normalized analytic functions and q-hypergeometric functions, where several interesting results were obtained (see [11,12]). In the year 2016, Wongsaijai and Sukantamala [17] published a paper, in which they generalized certain subclasses of starlike functions in a systematic way They made a very significant usage of the q-calculus basically in the context of Geometric Function Theory. Thereafter we will demonstrate three (presumably new) subclasses of the class Sq∗ of q-starlike functions associated with the Janowski functions

A Set of Definitions

Main Results and Their Demonstration

Analytic Functions with Negative Coefficients

Conclusions