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arXiv · 2011.02384

Outer functions and uniform integrability

Javad Mashreghi·Thomas Ransford
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Abstract

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

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Javad Mashreghi, Thomas Ransford. 2020-11-04. Outer functions and uniform integrability. https://doi.org/10.5186/aasfm.2018.4360

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