On the decay of singular inner functions

TL;DR

This paper demonstrates that the rate at which singular inner functions approach zero near the boundary of the unit disk can be made arbitrarily slow, highlighting variability in their boundary behavior.

Contribution

It proves that the convergence of singular inner functions to zero near the boundary can be arbitrarily slow, extending understanding of their boundary decay properties.

Findings

Convergence of singular inner functions to zero can be arbitrarily slow.

Boundary decay rates of singular inner functions are highly variable.

The result generalizes previous knowledge about their boundary behavior.

Abstract

It is known that, if $S(z)$ is a non-constant, singular inner function defined on the unit disk, then $\min_{|z|\le r}|S(z)|\to0$ as $r\to1^-$. We show that the convergence may be arbitrarily slow.