Sharp Coefficient Bounds for certain $q$-Starlike Functions

Abstract

Geometric function theory increasingly draws on $q$-calculus to model discrete and quantum-inspired phenomena. Motivated by this, the present paper introduces new subclasses of analytic functions: the class ${\mathcal{S}}_{\hat{I}{¾}_q}^{\ast }$ of $q$-starlike functions associated with the Ma-Minda function $\hat{I}{¾}_q(z)$, and its limiting classical counterpart ${\mathcal{S}}_{\hat{I}}^{\ast }¾$ associated with $\hat{I}¾(z)$, where $q\in (0,1)$. We systematically establish sharp coefficient estimates including the Fekete-Szegö, Hankel and Toeplitz determinants. We establish the sharpness of the $q$-coefficient estimates using a newly derived integral representation, which offers a more effective alternative to the conventional convolution-based extremal construction. It is further shown that all $q$-results reduce to their classical counterparts as $q\to 1^{-}$.

Earlier version has been modified by omitting certain results, title, keywords etc are modified