  The Hankel determinant $H_{2,1}(F_{f}/2)$ is defined as: \begin{align*}
  H_{2,1}(F_{f}/2):= \begin{vmatrix}
  \gamma_1 & \gamma_2
  \gamma_2 & \gamma_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 $|H_{2,1}(F_{f}/2)|\leq 1/16$ and
$|H_{2,1}(F_{f}/2)| \leq 23/3264$ for the logarithmic coefficients of starlike
and convex functions associated with lune.
