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 Google™ Scholar. Others By: Fernández, José L. - Pestana, Domingo
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 Title: Distortion of boundary sets under inner functions and applications Author(s): Fernández, José L.Pestana, Domingo Publisher: Indiana University, Department of Mathematics Issued date: 1992 Citation: Indiana University Mathematics Journal, 1992, vol. 41, n. 2, p. 439-448 URI: http://hdl.handle.net/10016/6557 ISSN: 0022-2518 DOI: 10.1512/iumj.1992.41.41025 Description: 10 pages, no figures.-- MSC2000 codes: 30C85, 30D50.MR#: MR1183352 (93k:30014)Zbl#: Zbl 0765.30011 Abstract: An inner function is a bounded holomorphic function from the unit disc $\Delta$ of the complex plane such that the radial boundary values have modulus 1 a.e. . If $E$ is a Borel subset of $\partial\Delta$ we also define $f(E)=\{e\sp{i\theta}/\lim\sb{r\to 1} f(re\sp{i\theta})$ exists and belongs to $E\}$. Let $M\sb \alpha$, $\text{cap}\sb \alpha$ and dim denote respectively the $\alpha$-dimensional content, $\alpha$- dimensional capacity and the Hausdorff dimension. In relation to the available results the authors in this paper prove that if $f$ is inner, $f(0)=0$, and $E$ is a Borel subset of $\partial\Delta$ then $M\sb \alpha(f\sp{-1}(E)) \geq C\sb \alpha M\sb \alpha(E)$ and for $0\leq\alpha<1$, $\text{cap}\sb \alpha(f\sp{-1}(E)) \geq C\sb \alpha \text{cap}\sb \alpha(E)$. An immediate consequence of course is $\dim(f\sp{-1}(E))\geq \dim E$. They also give examples to show that the inequalities cannot be reversed [source: Zentralblatt MATH]. Sponsor: The first author was supported in part by a grant from CICYT, Ministerio de Educación y Ciencia, Spain. Review: PeerReviewed Publisher version: http://www.iumj.indiana.edu/ Keywords: Inner functionBorel subsetHausdorff dimension Rights: © Indiana University Mathematics Journal Appears in Collections: DM - GAMA - Artículos de Revistas

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