On the angular limits of Bloch functions
A function f analytic in the unit disk \({\mathbb{D}}\) belongs to the space \({\mathcal B}\) of Bloch functions if \((1-| z|)| f'(z)|\) is bounded; it belongs to \({\mathcal B}_ 0\) if this tends to 0 if \(| z| \to 1\). In the paper, the power series expansion of f is divided into overlapping blocks such that these blocks are uniformly bounded in \({\mathbb{D}}\) if and only if \(f\in {\mathcal B}\). This process is used to assign to every \(f\in {\mathcal B}_ 0\) a ``larger'' function \(g\in {\mathcal B}\) such that, if \[ \sup_{0<r<1}| g(re^{it})| <\infty \] for some t then \(\lim_{r\to 1}f(re^{it})\neq \infty\) exists. A recent result of C. J. Bishop and P. W. Jones (preprint 1989) therefore implies that every function in \({\mathcal B}_ 0\) has finite angular limits on a set of Hausdorff dimension one.
scientific article; zbMATH DE number 4069273