Remarks on one-component inner functions

TL;DR

This paper investigates conditions under which certain inner functions are one-component, explores their derivatives in Hardy and Bergman spaces, and provides new criteria for Blaschke products and singular inner functions.

Contribution

It offers new sufficient conditions for Blaschke products and singular inner functions to be one-component, and characterizes derivatives of these functions in Hardy and Bergman spaces.

Findings

A Blaschke product with zeros in a Stolz domain can be a one-component inner function under certain conditions.

Sufficient conditions are established for atomic singular inner functions to be one-component.

The derivative of a one-component inner function belongs to Hardy space $H^p$ iff its second derivative is in Bergman space $A_{p-1}^p$.

Abstract

A one-component inner function $\Theta$ is an inner function whose level set $$\Omega_{\Theta}(\varepsilon)=\{z\in \mathbb{D}:|\Theta(z)|<\varepsilon\}$$ is connected for some $\varepsilon\in (0,1)$. We give a sufficient condition for a Blaschke product with zeros in a Stolz domain to be a one-component inner function. Moreover, a sufficient condition is obtained in the case of atomic singular inner functions. We study also derivatives of one-component inner functions in the Hardy and Bergman spaces. For instance, it is shown that, for $0<p<\infty$, the derivative of a one-component inner function $\Theta$ is a member of the Hardy space $H^p$ if and only if $\Theta''$ belongs to the Bergman space $A_{p-1}^p$, or equivalently $\Theta'\in A_{p-1}^{2p}$.