On radial behaviour and balanced Bloch functions (Q1970370)

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 \[ \sup_{z\in \Delta (\zeta, \rho)} \bigl(1-|z|^2\bigr) \bigl|f'(z)\bigr|\geq a\mu_f\bigl( |\zeta |\bigr), \quad\zeta\in D. \] A sufficient condition is given for a function represented by a power series with Hadamard gaps to be a balanced Bloch function, and a typical example of a function satisfying this condition is \(f(z)= \sum^\infty_{k=1} k^{-\gamma} z^{2^k}\), where \(0\leq\gamma <\infty\). Some properties of balanced Bloch functions are given. For example, if \(f\) is a balanced Bloch function with \(\int^1_0 {\mu_f(r) \over 1-r}dr= \infty\) and \(C\) is any curve in \(D\) ending on \(T\), then both \(f\) assumes every value in \({\mathcal C}\) infinitely often in \(D\) and \(\int_C|f'(z)|dz|=\infty\). As a consequence, for a balanced Bloch function \(f\), \(\int^1_0|f'(r\zeta)|dr<\infty\) holds either for all \(\zeta\in T\) or for none.