Two applications of Grunsky coefficients in the theory of univalent functions

TL;DR

This paper explores the use of Grunsky coefficients to address two problems in the theory of univalent functions: estimating the fourth logarithmic coefficient and bounding the difference between the fifth and fourth coefficients.

Contribution

It introduces a method based on Grunsky coefficients to analyze specific coefficient bounds within the class of univalent functions.

Findings

Estimated the fourth logarithmic coefficient for functions in  class 

Derived an upper bound for |a_5| - |a_4|

Provided new bounds using Grunsky coefficient techniques

Abstract

Let $\mathcal{S}$ denote the class of functions $f$ which are analytic and univalent in the unit disk ${\mathbb D}=\{z:|z|<1\}$ and normalized with $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. Using a method based on Grusky coefficients we study two problems over the class $\mathcal{S}$: estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference $|a_5|-|a_4|$.