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

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 \(f \in \mathcal{S}^*_\rho\).