Radial growth and variation of bounded analytic functions

Abstract

If a function f analytic in Δ = {z∈ℂ:|z|<1} has a nontangential limit as z→eiθ, then limr→1−(1−r)f′(reiθ)=0 [7, p. 181). It follows that this limit is zero for almost all θ for a number of classes of functions including the set H∞ of bounded analytic functions. In this paper we prove that this result for H∞ is sharp in a strong sense.