Sharp Bounds for $q$-Starlike Functions and Their Classical Counterparts

TL;DR

This paper introduces new subclasses of $q$-starlike and classical functions, providing sharp bounds and inequalities for their coefficients and determinants, advancing geometric function theory with $q$-calculus methods.

Contribution

It defines two new classes of analytic functions related to $q$-calculus and derives sharp bounds and inequalities for their coefficients and determinants.

Findings

Established sharp coefficient estimates including Fekete-Szeg"o, Kruskal, and Zalcman inequalities.

Derived sharp bounds for Hankel and Toeplitz determinants for the new classes.

Analyzed geometric properties of $q$-starlike functions and their classical counterparts.

Abstract

Geometric function theory increasingly draws on $q$-calculus to model discrete and quantum-inspired phenomena. Motivated by this, the present paper introduces two new subclasses of analytic functions: the class $\mathcal{S}^{*}_{\xi_q}$ of $q$-starlike functions associated with the Ma-Minda function $\xi_q(z)$, and its classical counterpart $\mathcal{S}^{*}_{\xi}$ associated with $\xi(z)$, where $q \in (0,1)$. We conduct a systematic investigation of the geometric properties of these function classes and establish sharp coefficient estimates, including Fekete-Szeg\"o, Kruskal, and Zalcman-type inequalities. Furthermore, we obtain sharp bounds of Hankel and Toeplitz determinants for both classes.