Results on Logarithmic Coefficients for the Class of Bounded Turning Functions

TL;DR

This paper investigates sharp bounds for logarithmic coefficients, their inverses, and related determinants in the class of bounded turning functions, advancing understanding of their coefficient inequalities and conjectures.

Contribution

It provides the first sharp bounds for logarithmic coefficients, inverse coefficients, and Hankel determinants specifically for bounded turning functions.

Findings

Sharp bounds for logarithmic coefficients established

Bounds for logarithmic inverse coefficients determined

Results support the generalized Zalcman conjecture in this class

Abstract

It is crucial to explore the sharp bounds of logarithmic coefficients and the Hankel determinant involving logarithmic coefficients as part of coefficient problems in various function classes. Our primary objective in this study is to determine the sharp bounds for logarithmic coefficients as well as logarithmic inverse coefficients of bounded analytic functions associated with a bean-shaped domain in the class $\mathcal{BT_\mathfrak{B}}$. For this class, we also establish the sharp bounds for the second Hankel determinant involving logarithmic coefficients as well as logarithmic inverse coefficients. In addition, we establish sharp bounds for the generalized Zalcman conjecture inequality and the moduli differences of logarithmic coefficients for the class $\mathcal{BT_\mathfrak{B}}$.