Snehal Pannu | Searcharxiv

arXiv2026

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 $q \to 1^{-}$.

math.CV↗

arXiv2024

On Coefficient problems for \textbf{$S^*_ρ$}

Logarithmic and inverse logarithmic coefficients play a crucial role in the theory of univalent functions. In this study, we focus on the class of starlike functions $\mathcal{S}^*_ρ$, defined as \[ \mathcal{S}^*_ρ= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec ρ(z), \; z \in \mathbb{D} \right\}, \] where $ρ(z) := 1 + \sinh^{-1}(z)$, which maps the unit disk $\mathbb{D}$ onto a petal-shaped domain. This investigation aims to establish bounds for the second Hankel and Toeplitz determinants, with their entries determined by the logarithmic coefficients of $f$ and its inverse $f^{-1}$, for functions $f \in \mathcal{S}^*_ρ$.

math.CV↗