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

TL;DR

This paper derives sharp bounds for the second Hankel determinant of logarithmic coefficients of inverse functions within certain classes of univalent functions, including starlike, convex, and bounded turning functions.

Contribution

It establishes the first sharp inequalities for the second Hankel determinant of inverse logarithmic coefficients in these specific classes.

Findings

Sharp bounds for starlike functions: 19/288

Sharp bounds for convex functions: 1/144

Sharp bounds for functions with bounded turning: 1/36

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} \Gamma_1 & \Gamma_2 \Gamma_2 & \Gamma_3 \end{vmatrix}=\Gamma_1\Gamma_3-\Gamma^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 $|H_{2,1}(F_{f^{-1}}/2)|\leq 19/288$, $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/144$, and $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$ for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order $1/2$, respectively.