Hankel and Toeplitz determinants of logarithmic coefficients of Inverse functions for certain classes of univalent functions

TL;DR

This paper derives sharp bounds for Hankel and Toeplitz determinants of logarithmic coefficients of inverse functions within certain classes of univalent functions, advancing understanding of their coefficient inequalities.

Contribution

It establishes the first sharp inequalities for these determinants for inverse functions of starlike and convex univalent functions, with supporting examples.

Findings

Sharp bounds for $H_{2,1}$ and $T_{2,1}$ determinants are proven.

Inequalities are validated with illustrative examples.

Results apply to classes of starlike and convex 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: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} \Gamma_1 & \Gamma_2 \Gamma_2 & \Gamma_3 \end{vmatrix} \;\;\mbox{and} \;\; T_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} \Gamma_1 & \Gamma_2 \Gamma_2 & \Gamma_1 \end{vmatrix} \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 $|H_{2,1}(F_{f^{-1}}/2)|\leq 1/4$, $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$, $|T_{2,1}(F_{f^{-1}}/2)|\leq 5/16$ and $|T_{2,1}(F_{f^{-1}}/2)|\leq 145/2304$ for the logarithmic coefficients of inverse functions for the classes starlike functions and convex functions with respect to symmetric points. In addition, our findings are substantiated further through the incorporation of illustrative examples, which support the strict inequality and lend credence to our conclusions.