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

Abstract: 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).