Multipoint Julia theorems

TL;DR

This paper introduces a multipoint Julia lemma using iterated hyperbolic difference quotients, providing new boundary behavior estimates for holomorphic self-maps of the unit disk, extending classical results.

Contribution

It develops a multipoint Julia lemma based on hyperbolic difference quotients, generalizing boundary derivative estimates and fixed point results in complex analysis.

Findings

Sharp lower bound for boundary angular derivatives.

Generalization of boundary estimates to multiple fixed points.

Systematic approach to higher order derivatives' influence on boundary behavior.

Abstract

Following ideas introduced by Beardon-Minda and by Baribeau-Rivard-Wegert in the context of the Schwarz-Pick lemma, we use the iterated hyperbolic difference quotients to prove a multipoint Julia lemma. As applications, we give a sharp estimate from below of the angular derivative at a boundary point, generalizing results due to Osserman, Mercer and others; and we prove a generalization to multiple fixed points of an interesting estimate due to Cowen and Pommerenke. These applications show that iterated hyperbolic difference quotients and multipoint Julia lemmas can be useful tools for exploring in a systematic way the influence of higher order derivatives on the boundary behaviour of holomorphic self-maps of the unit disk.