On certain applications of grunsky coefficients in the theory of univalent functions

TL;DR

This paper surveys how Grunsky coefficients can be used to derive sharp estimates for various coefficients and determinants in the theory of univalent functions, where no explicit analytical characterisation exists.

Contribution

It introduces a method based on Grunsky coefficients to obtain new estimates for coefficients and determinants in univalent function theory.

Findings

Derived sharp estimates for third and fourth logarithmic coefficients.

Provided bounds for second and third Hankel determinants.

Estimated coefficients of inverse functions and their differences.

Abstract

In this paper a survey is given of application of a method based on Grunsky coefficients for obtaining different estimates (some sharp) for the general class of univalent functions where no analytical characterisation exists. More precisely, estimates are given for the modulus of the third and the fourth logarithmic coefficients, for the modulus of the second and the third Hankel determinant for the general class of univalent functions, and for the modulus of some coefficients of the inverse function, and some coefficient differences.