On Coefficient problems for \textbf{0}

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 ${\mathcal{S}}_{\rho }^{\ast }$, defined as $${\mathcal{S}}_{\rho }^{\ast }=\{f\in \mathcal{A}:\frac{z{f}^{\mathrm{\prime }}(z)}{f(z)}\prec \rho (z),z\in \mathbb{D}\},$$ where $\rho (z):=1+{\sinh }^{-1}(z)$, 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 $f\in {\mathcal{S}}_{\rho }^{\ast }$.