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

The Hankel determinant H2,1Ff-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)$$\end{document} of logarithmic coefficients is defined asH2,1Ff-1/2:=Gamma 1 Gamma 2 Gamma 2 Gamma 3=Gamma 1 Gamma 3-Gamma 22,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right):=\left|\begin{array}{cc}{\Gamma }_{1}& {\Gamma }_{2}\\ {\Gamma }_{2}& {\Gamma }_{3}\end{array}\right|={\Gamma }_{1}{\Gamma }_{3}-{\Gamma }_{2}<^>{2},$$\end{document}where Gamma 1,Gamma 2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Gamma }_{1},{\Gamma }_{2},$$\end{document} and Gamma 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Gamma }_{3}$$\end{document} are the first, second, and third logarithmic coefficients of inverse functions belonging to the class S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}$$\end{document} of normalized univalent functions. In this paper, we establish sharp inequalities H2,1Ff-1/2 <= 19/288,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 19/288,$$\end{document}H2,1Ff-1/2 <= 1/144,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 1/144,$$\end{document} and H2,1Ff-1/2 <= 1/36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 1/36$$\end{document} for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively.