A characterization of random Bloch functions (Q1840553)

A Bloch function is an analytic function \(f\) on the unit disk \(D=\{z:|z|<1\}\) such that \(\sup_{z\in D}(1-|z|^2)|f'(z)|<\infty\). It is known that the set of all Bloch functions becomes a Banach space (called the Bloch space) with respect to the norm \(\|f\|_B=|f(0)|+\sup_{z\in D}(1-|z|^2)|f'(z)|\). In this paper the author obtains a necessary and sufficient condition for the complex sequence \(\{a_n\}\) with \(\sum|a_n|^2<\infty\) such that \(\sum_{n=1}^\infty\pm a_nz^n\) is a Bloch function for almost all choices of signs \(\pm\). Also the author answers to a question of \textit{J. M. Anderson, J. Clunie} and \textit{C. Pommerenke} [J. Reine Angew. Math. 270, 12-37 (1974; Zbl 0292.30030)].