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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 $Thetain A_{p-1}^{2p}$.
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
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In this paper, following Grothendieck {it Esquisse dun programme}, which was motivated by Belyis work, we study some properties of surfaces $X$ which are triangulated by (possibly ideal) isometric equilateral triangles of one of the spherical, euclid
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We study the geometry of the scale invariant Cassinian metric and prove sharp comparison inequalities between this metric and the hyperbolic metric in the case when the domain is either the unit ball or the upper half space. We also prove sharp disto