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

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 $\mathcal{S}^*_\rho$, defined as $$\mathcal{S}^*_\rho = \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \rho(z), \; z \in \mathbb{D} \right\},$$ 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 $f \in \mathcal{S}^*_\rho$.