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Contracting properties of bounded holomorphic functions. (English) Zbl 1469.30053

Summary: The classical Schwarz-Pick lemma implies that a holomorphic function of the open unit disk into itself gives a contraction with respect to the hyperbolic metric on the disk. In this article, we formulate a more general contracting property of a bounded holomorphic function in settings with respect to a conformal semimetric. During the study, it will become clear that a proper holomorphic mapping plays a significant role, and that a global result implies a local result. We conclude with a theorem for higher-dimensional spaces.

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
32H35 Proper holomorphic mappings, finiteness theorems
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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References:

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