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 ${\mathcal{S}}_{{\xi }_q}^{\ast }$ of $q$-starlike functions associated with the Ma-Minda function ${\xi }_q(z)$, and its limiting classical counterpart ${\mathcal{S}}_{\xi }^{\ast }$ associated with $\xi (z)$, 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 $q\to 1^{-}$.