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.

中文翻译：

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有界全纯函数的收缩性质

经典的Schwarz-Pick引理意味着开放单元磁盘本身的全纯函数相对于磁盘上的双曲度量会产生收缩。在本文中，我们针对共形半度量设置了有界全纯函数在设置中的更一般的收缩特性。在研究过程中，很明显，适当的全纯映射起着重要的作用，而全局结果则意味着局部结果。我们以高维空间定理作为结论。