On the integral means spectrum of univalent functions with quasconformal extensions

TL;DR

This paper proves that univalent functions with quasiconformal extensions have a strictly smaller integral means spectrum than the universal spectrum, answering a previously open question.

Contribution

It establishes a strict inequality for the integral means spectrum of such functions, advancing understanding in geometric function theory.

Findings

Integral means spectrum is strictly less for functions with quasiconformal extensions.

Provides an affirmative answer to a previously posed open question.

Enhances knowledge of the geometric properties of univalent functions.

Abstract

In this note we 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. This gives an affirmative answer to a question raised in our recent paper.