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arXiv · 2307.14365

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

Sanju Mandal·Molla Basir Ahamed
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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 $Γ_1, Γ_2,$ and $Γ_3$ 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.

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Sanju Mandal, Molla Basir Ahamed. 2023-07-25. Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions. https://arxiv.org/abs/2307.14365

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