Integral means spectrum functionals on Teichmuller spaces

TL;DR

This paper introduces and analyzes integral means spectrum functionals on Teichmüller spaces, establishing their continuity and comparing spectra of univalent functions with quasiconformal extensions.

Contribution

It defines IMS functionals on Teichmüller spaces, proves their continuity, and compares spectra for univalent functions with quasiconformal extensions.

Findings

IMS functionals are continuous on the closure of Teichmüller spaces

The spectrum of univalent functions with quasiconformal extensions is strictly less than the universal spectrum

New results are established for the Pre-Schwarzian derivative model

Abstract

In this paper we introduce and study the integral means spectrum (IMS) functionals on Teichm\"uller spaces. We show that the IMS functionals on the closure of the universal Teichm\"uller space and the universal asymptotic Teichm\"uller space are both continuous. During the proof, we consider the Pre-Schwarzian derivative model of universal asymptotic Teichm\"uller space and establish some new results for it. We also show that the integral means spectrum of any univalent function admitting a quasiconformal extension to the extended complex plane is strictly less than the universal integral means spectrum.