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Register a new userRemarks on one-component inner functions
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Authors
Atte Reijonen
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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 $Thetain A_{p-1}^{2p}$.
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We present a characterization of one-component inner functions in terms of the location of their zeros and their associated singular measure. As consequence we answer several questions posed by J. Cima and R. Mortini. In particular we prove that for any inner function $Theta$ whose singular set has measure zero, one can find a Blaschke product $B$ such that $Theta B$ is one-component. We also obtain a characterization of one-component singular inner functions which is used to produce examples of discrete and continuous one-component singular inner functions.
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