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: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_3 \end{vmatrix} \;\;\mbox{and} \;\; T_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_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 $|H_{2,1}(F_{f^{-1}}/2)|\le 1/4$, $|H_{2,1}(F_{f^{-1}}/2)|\le 1/36$, $|T_{2,1}(F_{f^{-1}}/2)|\le 5/16$ and $|T_{2,1}(F_{f^{-1}}/2)|\le 145/2304$ for the logarithmic coefficients of inverse functions for the classes 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.

15 pages. arXiv admin note: substantial text overlap with arXiv:2305.12500, arXiv:2307.14365