A characterization of one‐component inner functions

Abstract

We present a characterization of one-component inner functions in terms of the location of their zeros and their associated singular measure. As consequence we answer several questions posed by Cima and Mortini. In particular, we prove that for any inner function Θ whose singular set has measure zero, one can find a Blaschke product B such that BΘ is one-component. We also obtain a characterization of one-component singular inner functions which is used to produce examples of discrete and continuous one-component singular inner functions.

Blaschke Products, Level Sets, and Crouzeix's Conjecture

“…For example, recall that an inner function Θ is a one-component inner function if there exists an ε > 0 such that Ω Θ ε is connected. These were introduced by B. Cohn [11] and in the interim have been heavily studied, see [1,9,10,42]. Cohn introduced this class because he was able to characterize the Carleson measures for the model spaces H 2 ΘH 2 , under the assumption that Θ was a one-component inner function.…”