On the growth and coefficients of analytic functions (Q1059737)

Suppose that \(\Omega\) is a plane domain such that \({\mathbb{C}}| \Omega\) is infinite. Hayman, Pommerenke, and Patterson proved that the universal covering map F from the unit disk \(\Delta\) onto \(\Omega\) satisfies \(S(r,F)=o(1-r)^{-1}\), where S(r,f) denotes the spherical area counting multiplicities of the image of F under \(| z| \leq r\). They conjectured the same bound would hold for any analytic \(f: \Delta\) \(\to \Omega\). Here the author, via an exceedingly ingenious iterative construction, shows the HPP conjecture is false. In fact, as long as \(cap({\mathbb{C}}| \Omega)=0\) there exists \(f: \Delta\) \(\to \Omega\) such that \(S(r,f)\neq o(1-r)^{-1}.\) This gives a new characterization of sets of capacity zero. The same construction also settles negatively a longstanding question about coefficients of Bloch functions: If \(cap({\mathbb{C}}| \Omega)=0\) there exists \(f: \Delta\) \(\to \Omega\) for which \(a_ n\neq O(1).\) The author has used a related construction to settle affirmatively a question of Hayman about the relation between Nevanlinna's proximity function and sets of given Hausdorff dimension [J. Lond. Math. Soc., II. Ser. 30, 79-86 (1984; Zbl 0561.30024)].