Complex exponential integral means spectra of univalent functions and the Brennan conjecture

TL;DR

This paper studies the spectra of univalent functions, proving continuity of integral means spectra on various spaces, establishing bounds related to the Brennan conjecture, and fully determining spectra for rational functions.

Contribution

It introduces a new direct approach to prove spectrum continuity, extends results to universal Teichmüller spaces, and confirms the Brennan conjecture for univalent rational functions.

Findings

All IMS functionals are continuous on key Teichmüller spaces.

The spectrum of univalent functions with quasiconformal extensions is below the universal spectrum.

The Brennan conjecture holds for univalent rational functions.

Abstract

In this paper we investigate the complex exponential integral means spectra of univalent functions in the unit disk. We show that all integral means spectrum (IMS) functionals for complex exponents on the universal Teichm\"uller space, the closure of the universal Teichm\"uller curve, and the universal asymptotic Teichm\"uller space are continuous. We also show that the complex exponential 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. These extend some related results in our recent work \cite{Jin}. Here we employ a different and more direct approach to prove the continuity of the IMS functional on the universal asymptotic Teichm\"uller space. Additionally, we completely determine the integral means spectra of all univalent rational functions in the unit disk. As a consequence, we show that the Brennan conjecture is true for this class of univalent functions. Finally, we present some remarks and raise some problems and conjectures regarding IMS functionals on Teichm\"uller spaces, univalent rational functions, and a multiplication operator whose norm is closely related to the Brennan conjecture.