The growth of bounded and related functions

Abstract

In this paper, an answer is supplied to the following question of Rozenblum: ‘How fast can a bounded analytic function in the unit disc tend to zero outside an appropriate exceptional set’? The answer is obtained first for Blaschke products and is then extended to the Nevanlinna class of functions of bounded characteristic, that is, meromorphic functions which are the ratio of two bounded functions, by the representation of such functions as the ratio of two Blaschke products multiplied by an exponential term. For Blaschke products B ( z ) , it is shown that ( 1 − | z | ) log | B ( z ) | ⟶ 0 , as | z | → 1 for z outside an exceptional F-set. This is defined in the introduction as the union of a suitable set of discs lying in the unit disc. It is also shown that both the rate of growth implied by the above limit and the size of the exceptional set characterised by the concept of an F-set are essentially sharp.