Contracting properties of bounded holomorphic functions

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.