Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for Certain Classes of Univalent Functions

Abstract

The Hankel and Toeplitz determinants \(H_{2,1}(F_{f^{-1}}/2)\) and \(T_{2,1}(F_{f^{-1}}/2)\) are defined as: \(H_{2,1}(F_{f^{-1}}/2):=\Gamma _{1}\Gamma _{3} -\Gamma ^2_{2}\) and \(T_{2,1}(F_{f^{-1}}/2):=\Gamma ^2_{1}-\Gamma ^2_{2}\), 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 article, we establish sharp inequalities \(|H_{2,1}(F_{f^{-1}}/2)|\le 1/4\), \(|H_{2,1}(F_{f^{-1}}/2)| \le 1/36\), \(|T_{2,1}(F_{f^{-1}}/2)|\le 5/16\) and \(|T_{2,1}(F_{f^{-1}}/2)|\le 145/2304\) for the logarithmic coefficients of inverse functions for the classes of starlike functions and convex functions with respect to symmetric points. The results show an invariance property of the second Hankel determinants \(H_{2,1}(F_{f}/2)\) and \(H_{2,1}(F_{f^{-1}}/2)\) of logarithmic coefficients for the classes \(\mathcal {S}^*_S\) and \(\mathcal {K}_S\). Moreover, we exhibit examples showing that the strict inequality in the main results hold.