Outer functions and uniform integrability

Abstract

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