Sharp bounds for second Hankel determinant of logarithmic coefficients for certain classes of univalent functions

Abstract

The Hankel determinant $H_{2,2}(F_f/2)$ is defined as: \begin{align*} H_{2,2}(F_{f}/2):= \begin{vmatrix} γ_2 & γ_3 γ_3 & γ_4 \end{vmatrix}, \end{align*} where ${\gamma }_2,{\gamma }_3,$ and ${\gamma }_4$ are the second, third, and fourth logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,2}(F_f/2)|\le (1272+113\sqrt{678})/32856$ and $|H_{2,2}(F_f/2)|\le 13/1080$ for the logarithmic coefficients of starlike and convex functions with respect to symmetric points. Moreover, we provide examples that demonstrate the strict inequality holds.

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