A one-component inner function $Theta$ is an inner function whose level set $$Omega_{Theta}(varepsilon)={zin mathbb{D}:|Theta(z)|<varepsilon}$$ is connected for some $varepsilonin (0,1)$. 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 $Thetain A_{p-1}^{2p}$.
تحميل البحث