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

Abstract

The Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic coefficients is defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_3 \end{vmatrix}=Γ_1Γ_3-Γ^2_2, \end{align*} 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 $|H_{2,1}(F_{f^{-1}}/2)|\le 19/288$, $|H_{2,1}(F_{f^{-1}}/2)|\le 1/144$, and $|H_{2,1}(F_{f^{-1}}/2)|\le 1/36$ for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order $1/2$, respectively.

14. arXiv admin note: substantial text overlap with arXiv:2307.02741