Strengthened Grunsky and Milin inequalities

TL;DR

This paper enhances classical Grunsky inequalities for univalent functions and extends these results to quasiconformal disks, broadening their applicability and providing new theoretical insights.

Contribution

The paper develops a quasiconformal variant of Grunsky inequalities, extending classical results to a wider class of functions and surfaces.

Findings

Improved bounds for classical Grunsky inequalities.

Extension of inequalities to quasiconformal disks.

Multiple applications demonstrated.

Abstract

The method of Grunsky inequalities has many applications and has been extended in many directions, even to bordered Riemann surfaces. However, unlike the case of functions univalent in the disk, a quasiconformal variant of this theory has not been developed so far. In this paper, we essentially improve the basic facts concerning the classical Grunsky inequalities for univalent functions on the disk and extend these results to arbitrary quasiconformal disks. Several applications are given.