An Application of Jackson's $(p, q)$-Derivative to a Subclass of Starlike Functions with Negative Coefficients

TL;DR

This paper introduces a new subclass of starlike functions with negative coefficients using Jackson's $(p, q)$-derivative, deriving various properties, inequalities, and radii that generalize previous results in geometric function theory.

Contribution

It defines a novel subclass of starlike functions with negative coefficients using a differential operator and establishes new inequalities, radii, and properties that extend earlier research.

Findings

Coefficient inequalities established

Growth and distortion theorems proved

Radii of starlikeness and convexity determined

Abstract

In this paper, we introduce and investigate the subclass $\mathcal{P}_{p,q}^{\xi ,\kappa}(\tau, \eta)$ of starlike functions with negative coefficients by using the differential operator $\Upsilon_{\tau ,p,q}^{\xi ,\kappa}$. Coefficient inequalities, growth and distortion theorems, closure theorems, and some properties of several functions belonging to this class are obtained. We also determine the radii of close-to-convexity, starlikeness, and convexity for functions belonging to the class $\mathcal{P}_{p,q}^{\xi ,\kappa}(\tau, \eta)$. Furthermore, we obtain the integral means inequality and neighborhood results for functions belonging to the class $\mathcal{P}_{p,q}^{\xi ,\kappa}(\tau, \eta)$. The results presented in this paper generalize or improve those in related works of several earlier authors.