Distortion of boundary sets under inner functions and applications

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].