Outer functions and uniform integrability

TL;DR

This paper demonstrates that compositions involving outer functions and holomorphic self-maps of the disk produce uniformly integrable functions, providing a straightforward proof that outer functions remain outer under composition.

Contribution

It establishes uniform integrability of certain logarithmic functions related to outer functions and proves that outer functions are preserved under composition with holomorphic maps.

Findings

The set of log-modulus functions is uniformly integrable on the unit circle.

Outer functions composed with holomorphic self-maps remain outer functions.

Provides a simple proof of the composition property of outer functions.

Abstract

We show that, if $f$ is an outer function and $a\in[0,1)$, then the set of functions $\{\log |(f\circ\psi)^*|: \psi:\mathcal{D}\to\mathcal{D} \text{ holomorphic}, |\psi(0)|\le a\}$ is uniformly integrable on the unit circle. As an application, we derive a simple proof of the fact that, if $f$ is outer and $\phi:\mathcal{D}\to\mathcal{D}$ is holomorphic, then $f\circ\phi$ is outer.