Some application of Grunsky coefficients in the theory of univalent functions

TL;DR

This paper explores the application of Grunsky coefficients to derive bounds on various coefficients and determinants in the class of univalent functions, advancing understanding of their structural properties.

Contribution

It introduces a method based on Grunsky coefficients to analyze and establish bounds for specific coefficients and determinants in univalent function theory.

Findings

Derived upper bounds for the generalized Zalcman conjecture case

Established bounds for the third logarithmic coefficient

Provided estimates for the second Hankel determinant of logarithmic coefficients

Abstract

Let function $f$ be normalized, analytic and univalent in the unit disk ${\mathbb D}=\{z:|z|<1\}$ and $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. Using a method based on Grusky coefficients we study several problems over that class of univalent functions: upper bounds of the special case of the generalised Zalcman conjecture $|a_2a_3-a_4|$, of the third logarithmic coefficient, and of the second Hankel determinant for the logarithmic coefficients.