One-component inner functions II

Abstract

We continue our study of the set ${\mathfrak{I}}_c$ of inner functions $u$ in $H^{\infty }$ with the property that there is $\eta \in ]0,1[$ such that the level set ${\Omega }_u(\eta ):=\{z\in \mathbb{D}:|u(z)|<\eta \}$ is connected. These functions are called one-component inner functions. They play an important role in function theory and operator theory. Here we show that the composition of two one-component inner functions is again in ${\mathfrak{I}}_c$. We also give conditions under which a factor of one-component inner function belongs to ${\mathfrak{I}}_c$.

9 pages, 1 figure