Sharp Coefficient Bounds for certain $q$-Starlike Functions

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 $\mathcal{S}^{*}_{\xi_q}$ of $q$-starlike functions associated with the Ma-Minda function $\xi_q(z)$, and its limiting classical counterpart $\mathcal{S}^{*}_{\xi}$ associated with $\xi(z)$, where $q \in (0,1)$. We systematically establish sharp coefficient estimates including the Fekete-Szeg\"{o}, 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 $q \to 1^{-}$.