Brennan's conjecture holds for semigroups of holomorphic functions

TL;DR

This paper provides a concise proof of Brennan's conjecture specifically for continuous semigroups of holomorphic functions, utilizing classical complex analysis techniques and recent advances in weights and spectra of operators.

Contribution

It offers the first proof of Brennan's conjecture for a specific class of holomorphic function semigroups, combining classical and modern methods.

Findings

Brennan's conjecture is confirmed for continuous semigroups of holomorphic functions.

The proof integrates classical complex analysis with recent operator theory results.

The approach may inspire further research in related function theory problems.

Abstract

In the present note, we give a short proof of Brennan's conjecture in the special case of continuous semigroups of holomorphic functions. We apply classical techniques of complex analysis in conjunction with recent results on B\'{e}koll\'{e}-Bonami weights and spectra of integration operators.