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 $\{\log |(f\circ\psi)^*|: \psi:\mathcal{D}\to\mathcal{D} \text{ holomorphic}, |\psi(0)|\le a\}$ 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.