  The Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic coefficients is
defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} \Gamma_1 &
\Gamma_2 \Gamma_2 & \Gamma_3 \end{vmatrix}=\Gamma_1\Gamma_3-\Gamma^2_2,
\end{align*} where $\Gamma_1, \Gamma_2,$ and $\Gamma_3$ are the first, second
and third logarithmic coefficients of inverse functions belonging to the class
$\mathcal{S}$ of normalized univalent functions. In this article, we establish
sharp inequalities $|H_{2,1}(F_{f^{-1}}/2)|\leq 19/288$,
$|H_{2,1}(F_{f^{-1}}/2)| \leq 1/144$, and $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$
for the logarithmic coefficients of inverse functions, considering starlike and
convex functions, as well as functions with bounded turning of order $1/2$,
respectively.
