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Multipoint Julia theorems. (English) Zbl 1492.30056

The author uses the iterated hyperbolic difference quotients to prove a multipoint Julia lemma in the context of the Schwarz-Pick lemma. This follows some ideas from Beardon-Minda, Baribeau-Rivard-Wegert. As applications, the author gives a sharp estimate from below of the angular derivative at a boundary point, generalizing the results due to Osserman, Mercer and others; and also proves 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.

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30E25 Boundary value problems in the complex plane
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30J10 Blaschke products
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References:

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