Sharp Estimates of Logarithmic Coefficients for a Certain Class of Starlike Functions

TL;DR

This paper derives sharp bounds for logarithmic coefficients and Hankel determinants for a class of starlike functions related to hyperbolic cosine, including inverse functions, with extremal functions confirming the results.

Contribution

It provides the first sharp bounds for logarithmic coefficients and Hankel determinants for the class al{S}_{ch}^* and its inverse, extending the understanding of extremal properties.

Findings

Sharp upper bounds for al{ ext{logarithmic coefficients}} al{ abla}_n for n=1,2,3

Precise bounds for the second Hankel determinant H_{2,1} for the class and its inverse

Extremal functions constructed to verify the sharpness of all inequalities

Abstract

In this article, we investigate the extremal properties of logarithmic coefficients for the class $\mathcal{S}_{ch}^*$ of starlike functions associated with the hyperbolic cosine function. We establish the sharp upper bounds for the initial logarithmic coefficients $\gamma_n$ for $n=1, 2, 3$, and determine the precise bound for the second Hankel determinant $H_{2,1}(F_f/2)$ within this class. Furthermore, we extend our analysis to the inverse functions, deriving sharp estimates for the logarithmic inverse coefficients and the corresponding second Hankel determinant $|H_{2,1}(F_{f^{-1}}/2)|$. Additionally, we provide sharp bounds for the moduli differences of both logarithmic and inverse logarithmic coefficients. The sharpness of all obtained inequalities is verified through the construction of specific extremal functions.