On radial behaviour and balanced Bloch functions
type: Item
instance of: scholarly article
title: On radial behaviour and balanced Bloch functions
zbMATH Open document ID: 0948.30039
DOI: 10.4171/RMI/261
author: Juan Jesús Donaire
author: Christian Pommerenke
published in: Revista Matemática Iberoamericana
publication date: 2000-11-19T00:00:00Z
full work available at URL: https://eudml.org/doc/39582
review text: This paper deals with two aspects of Bloch functions: a generalization of a theorem on the boundary behavior of ``bad'' Bloch functions, and some properties of balanced Bloch functions. Let \({\mathcal C}\) denote the complex plane. For a function \(f\) analytic in the unit disc \(D\), let \(\|f\|_{\mathcal B}= \sup\{(1-|z|^2) |f'(z)|: 0\leq|z|<1\}\), and let \({\mathcal B}=\{f\) analytic in \(D:\|f\|_{\mathcal B}< \infty\}\), the space of Bloch functions. For a set \(E\), let \(\dim E\) denote the Hausdorff dimension of \(E\). The authors prove the following. Theorem 1.1. Let \(f\in{\mathcal B}\), \(\|f\|_B\leq 1\), and \(f(0)=0\). Let \(0\in G\subset{\mathcal C}\) such that for almost all \(\zeta\in\partial D\), \(\lim_{r\to 1}f(r\zeta)\) either lies in \({\mathcal C}-G\) or does not exist, and let \(\Gamma\) be any half open curve in \(G\) starting at 0. If \[ C_1<R<\text{dist}(0,\partial G),\quad\text{dist} (\Gamma, \partial G)\geq 2R, \] then there exists a set \(E_\Gamma\subset T\) with \(\dim E_\Gamma\geq 1-{C_2\over R}\) such that, for \(\zeta\in E_\Gamma\), there exists a parametrization \(\gamma_\zeta(r)\), \(0\leq r<1\), of \(\Gamma\) with \(\gamma_\zeta(0)=0\) and such that \(|f(r\zeta)- \gamma_\zeta (r)|\leq 2R\), \(0\leq r<1\). Here, \(C_1\) and \(C_2\) are absolute constants. This result generalizes a result of \textit{S. Rohde} [J. Lond. Math. Soc., II. Ser. 48, 488-499 (1993; Zbl 0792.30023)], who proved this for \(G={\mathcal C}\). An application of Theorem 1.1 to conformal mappings is given. For \(0\leq r<1\) and \(f\in{\mathcal B}\), let \(\mu_f(r)= \sup_{r\leq |z|>1}(1-|z|^2) |f'(z)|\). For \(\zeta\in D\) and \(\rho>0\), let \(\Delta(\zeta,\rho)\) denote the non-Euclidean disk with center as \(\zeta\) and radius \(\rho\). The function \(f\in{\mathcal B}\) is called a balanced Bloch function if there exists a constant \(a>0\) and a \(\rho<\infty\) such that \[ \sup_{z\in \Delta (\zeta, \rho)} \bigl(1-|z|^2\bigr) \bigl|f'(z)\bigr|\geq a\mu_f\bigl(…
Mathematics Subject Classification ID: 30D40
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 1419784
zbMATH Keywords: Bloch functions
zbMATH Keywords: conformal mappings
reviewed by: Peter Lappan
MaRDI profile type: Publication
OpenAlex ID: W2076803155
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description: scientific article; zbMATH DE number 1419784