Sharp bounds for second Hankel determinant of logarithmic coefficients for certain classes of univalent functions

The Hankel determinant $H_{2,2}(F_f\mathrm{/}2)$ is defined as: \begin{align*} H_{2,2}(F_{f}/2):= \begin{vmatrix} γ_2 & γ_3 γ_3 & γ_4 \end{vmatrix}, \end{align*} where ${\gamma }_2,{\gamma }_3,$ and ${\gamma }_4$ are the second, third, and fourth logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $\mathrm{\mid }{H}_{2,2}(F_f\mathrm{/}2)\mathrm{\mid }\le (1272+113\sqrt{678})\mathrm{/}32856$ and $\mathrm{\mid }{H}_{2,2}(F_f\mathrm{/}2)\mathrm{\mid }\le 13\mathrm{/}1080$ for the logarithmic coefficients of starlike and convex functions with respect to symmetric points. Moreover, we provide examples that demonstrate the strict inequality holds.