Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions

Abstract

The Hankel determinant \({H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\) of logarithmic coefficients is defined as

\({H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right):=\left|\begin{array}{cc}{\Gamma }_{1}& {\Gamma }_{2}\\ {\Gamma }_{2}& {\Gamma }_{3}\end{array}\right|={\Gamma }_{1}{\Gamma }_{3}-{\Gamma }_{2}^{2},\)

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 \(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 19/288,\) \(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 1/144,\) and \(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 1/36\) for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively.