The Julia-Wolff-Carathéodory theorem(s)

As satisfying as it is from several points of view, this theorem leaves open the question of what happens at a specific point σ0 ∈ ∂∆. Of course, to get a sensible statement one needs to make some assumptions on the function f . In 1920, Julia ([Ju]) identified the right hypotheses, showing how to get the existence of the non-tangential limit at a given boundary point using Schwarz’s lemma. But the real breakthrough is due to Wolff ([W]) in 1926 and Carathéodory ([C]) in 1929, who proved that under Julia’s hypotheses the derivative too admits non-tangential limit at the specified boundary point. Their results are collected in the following statement, the Julia-Wolff-Carathéodory theorem: