One-component inner functions II

TL;DR

This paper investigates the properties of one-component inner functions, demonstrating that their composition remains within the class and establishing conditions for factors to also belong to this class, thus advancing understanding in function and operator theory.

Contribution

It proves that the composition of two one-component inner functions is again one-component and provides conditions for factors to be in the class.

Findings

Composition of two one-component inner functions remains in the class.

Conditions identified for factors of one-component inner functions to also be in the class.

Highlights the importance of these functions in function and operator theory.

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 $\mathfrak I_c$.