On Blaschke products, Bloch functions and normal functions

Abstract

We prove that if G is an analytic function in the unit disc such that G(z)→∞, as z→1, and B is an infinite Blaschke product whose sequence of zeros is contained in a Stolz angle with vertex at 1 then the function f=B⋅G is not a normal function. We prove also some results on the asymptotic cluster set of a thin Blaschke product with positive zeros which are related with the question of the existence of non-normal outer functions with restricted mean growth of the derivative.