Remarks on one-component inner functions

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 $\Theta'\in A_{p-1}^{2p}$.