Remarks on one-component inner functions

A one-component inner function $\mathrm{\Theta }$ is an inner function whose level set $${\mathrm{\Omega }}_{\mathrm{\Theta }}(\epsilon )=\{z\in \mathbb{D}:\mathrm{\mid }\mathrm{\Theta }(z)\mathrm{\mid }<\epsilon \}$$ is connected for some $\epsilon \in (0,1)$. 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 $0<p<\mathrm{\infty }$, the derivative of a one-component inner function $\mathrm{\Theta }$ is a member of the Hardy space $H^p$ if and only if ${\mathrm{\Theta }}^{\mathrm{\prime }\mathrm{\prime }}$ belongs to the Bergman space $A_{p-1}^p$, or equivalently ${\mathrm{\Theta }}^{\mathrm{\prime }}\in A_{p-1}^{2p}$.