Quasiconformal deformations preserving Hilbert's norm and their applications

TL;DR

This paper develops quasiconformal deformations that preserve Hilbert norms of holomorphic functions, enabling new estimates of Taylor coefficients and solving longstanding problems in complex analysis.

Contribution

It introduces a novel method of quasiconformal deformations preserving Hilbert norms, advancing the understanding of coefficient estimation and solving classical problems.

Findings

Constructed quasiconformal deformations preserving Hilbert norms.

Provided a nearly complete solution to the Hummel-Scheinberg-Zalcman problem.

Enhanced methods for estimating Taylor coefficients in complex analysis.

Abstract

This paper focuses on estimating the Taylor coefficients for Hilbert spaces of holomorphic functions on the disk using intrinsic features of univalent functions and of Teichmuller spaces. Estimating these coefficients has a long history but still remains an important problem in many geometric and physical applications of complex analysis. We construct quasiconformal deformations of holomorphic functions preserving their Hilbert norm. Such deformations play a crucial role in this subject. Among their applications, a rather complete solution of an old Hummel-Scheinberg-Zalcman problem is obtained.