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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.
Cite this article
Manabu Ito, Contracting properties of bounded holomorphic functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 1, pp. 135–147DOI 10.4171/RLM/930