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arXiv · 2305.12500

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

Sanju Mandal·Partha Pratim Roy·Molla Basir Ahamed
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Abstract

The Hankel determinant $H_{2,2}(F_{f}/2)$ is defined as: \begin{align*} H_{2,2}(F_{f}/2):= \begin{vmatrix} \gamma_2 & \gamma_3 \gamma_3 & \gamma_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 $|H_{2,2}(F_{f}/2)|\leq (1272 + 113\sqrt{678})/32856$ and $|H_{2,2}(F_{f}/2)| \leq 13/1080$ 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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Sanju Mandal, Partha Pratim Roy, Molla Basir Ahamed. 2023-05-21. Sharp bounds for second Hankel determinant of logarithmic coefficients for certain classes of univalent functions. https://arxiv.org/abs/2305.12500

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