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