The third Hankel determinant for inverse coefficients of starlike functions of order 1/2

Abstract

Let f be a starlike function of order 1/2 and let f-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{-1}$$\end{document} denote its local inverse. In this paper, we establish the sharp bound for the third Hankel determinant associated with the inverse coefficients of f-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{-1}$$\end{document}. Specifically, we prove that |H3(1)(f-1)|≤1/9,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|H_{3}(1)(f^{-1})|\le 1/9,$$\end{document} and show that the estimate is sharp. Since the sharp bound |H3(1)(f)|≤1/9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|H_{3}(1)(f)|\le 1/9$$\end{document} is already known for the class S∗(1/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}^{*}(1/2)$$\end{document}, our result yields the invariance supf∈S∗(1/2)|H3(1)(f)|=19=supf∈S∗(1/2)|H3(1)(f-1)|.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sup _{f\in \mathcal {S}^{*}(1/2)}|H_{3}(1)(f)|=\frac{1}{9}=\sup _{f\in \mathcal {S}^{*}(1/2)}|H_{3}(1)(f^{-1})|. $$\end{document}This provides the first invariance result for the third Hankel determinant between a starlike function of order 1/2 and its inverse.