A convex function f of order \(\alpha \in [0,1)\) is defined by \(\begin{aligned} \Re \left( 1+\frac{zf^{\prime\prime}(z)}{f'^{\prime}(z)}\right) >\alpha \quad \text {for } z \in \mathbb {D}, \end{aligned}\) where \(\mathbb {D}\) denotes the unit disk. The second Hankel determinant \(H_{2,1}\left( F_{f^{-1}} / 2\right) \) of logarithmic coefficients of inverse functions is defined as \(\begin{aligned} H_{2,1}\left( F_{f^{-1}} / 2\right) = \Gamma _1 \Gamma _3 - \Gamma _2^2, \end{aligned}\) 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 paper, we establish sharp inequalities \(\begin{aligned} |H_{2,1}(F_{f^{-1}}/2)|\le {\left\{ \begin{array}{ll}\frac{69\alpha ^2-108\alpha +48}{1584}, & \alpha \in [0,\frac{6-\sqrt{3}}{11}),\\ \frac{(1-\alpha )^2}{36}, & \alpha \in [\frac{6-\sqrt{3}}{11},\frac{4}{5}],\\ \frac{(1-\alpha )^2(69\alpha ^2-48\alpha +12)}{144(11\alpha ^2-2\alpha -1)}, & \alpha \in (\frac{4}{5},1) \end{array}\right. } \end{aligned}\) for the logarithmic coefficients of inverse convex function of order \(\alpha \) . The derived bound for \(\alpha \in [0,1)\) generalizes the results previously obtained by Allu and Shaji (Bull Austr Math Soc, 2025. https://doi.org/10.1017/S0004972724000200) and Mandal and Ahamed (Lith Math J 64:67–79, 2024).
