Multipoint Julia theorems

Following ideas introduced by Beardon–Minda and by Baribeau–Rivard–Wegert in the context of the Schwarz–Pick lemma, we use the iterated hyperbolic di¤erence 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 di¤erence 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.