Second Hankel determinant of Logarithmic coefficients for Starlike and Convex functions associated with lune

The Hankel determinant $H_{2,1}(F_f\mathrm{/}2)$ is defined as: \begin{align*} H_{2,1}(F_{f}/2):= \begin{vmatrix} γ_1 & γ_2 γ_2 & γ_3 \end{vmatrix}, \end{align*} where ${\gamma }_1,{\gamma }_2,$ and ${\gamma }_3$ are the first, second and third logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $\mathrm{\mid }{H}_{2,1}(F_f\mathrm{/}2)\mathrm{\mid }\le 1\mathrm{/}16$ and $\mathrm{\mid }{H}_{2,1}(F_f\mathrm{/}2)\mathrm{\mid }\le 23\mathrm{/}3264$ for the logarithmic coefficients of starlike and convex functions associated with lune.