The Angular Derivative

Abstract

At the end of the last chapter we managed to coax a fair amount of geometric intuition out of the condition

$$ \mathop {\lim }\limits_{\left| z \right| \to 1 - } \frac{{1 - \left| {\varphi \left( z \right)} \right|}}{{1 - \left| z \right|}} = \infty $$

(1)

which characterizes compactness for univalently induced composition operators. Because this condition involves the limit of a difference quotient, one might suspect that its real meaning is wrapped up in the boundary behavior of the derivative of φ. This is exactly what happens: we will see shortly that condition (1) is the hypothesis of the classical Julia-Carathéodory Theorem, which characterizes the existence of the “angular derivative” of φ at points of ∂U, and provides a compelling geometric interpretation of (1) in terms of “conformality at the boundary.” After discussing its connection with the compactness problem, we present a proof of the Julia-Carathéodory Theorem that emphasizes the role of hyperbolic geometry. The following terminology describes the limiting behavior involved in this circle of ideas.