Second Hankel determinant of Logarithmic coefficients for Starlike and Convex functions associated with lune

TL;DR

This paper derives sharp bounds for the second Hankel determinant of logarithmic coefficients for starlike and convex functions related to lune, advancing understanding of their coefficient structure.

Contribution

It establishes the first sharp inequalities for the second Hankel determinant of logarithmic coefficients for these specific function classes.

Findings

Sharp inequality |H_{2,1}(F_{f}/2)| ≤ 1/16 for starlike functions

Sharp inequality |H_{2,1}(F_{f}/2)| ≤ 23/3264 for convex functions

Provides new bounds for logarithmic coefficients in geometric function theory

Abstract

The Hankel determinant $H_{2,1}(F_{f}/2)$ is defined as: \begin{align*} H_{2,1}(F_{f}/2):= \begin{vmatrix} \gamma_1 & \gamma_2 \gamma_2 & \gamma_3 \end{vmatrix}, \end{align*} where $\gamma_1, \gamma_2,$ and $\gamma_3$ are the first, second and third logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/16$ and $|H_{2,1}(F_{f}/2)| \leq 23/3264$ for the logarithmic coefficients of starlike and convex functions associated with lune.