Let \(\mathcal {S}\) denote the class of univalent functions in the unit disk \(\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}\) with the form \(f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}\) . The logarithmic inverse coefficients \(\Gamma _{n}\) of \(f\in \mathcal {S}\) are defined by \(F_{f^{-1}}(w):=\log \left( f^{-1}(w)/w\right) =2\sum _{n=1}^{\infty }\Gamma _{n}w^{n}\) valid for some disk \(|w|\le r_{0}(f)\) . The second Hankel determinant of logarithmic inverse coefficients is defined by \(\begin{aligned} H_{2}\left( 2\right) \left( F_{f^{-1}}/2\right) =\Gamma _{2}\Gamma _{4}-\Gamma _{3}^{2}. \end{aligned}\) In this paper, we obtain sharp upper bound of the second Hankel determinant \(H_{2}\left( 2\right) \left( F_{f^{-1}}/2\right) \) of the logarithmic inverse coefficients of starlike and bounded turning functions.
