Inequalities of Schwarz–Pick type with branch points and the pseudo-hyperbolic distance

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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. As such, differential forms of the Schwarz–Pick lemma have been recently discussed in consideration of branch points of a holomorphic function. We in this article demonstrate that the pseudo-hyperbolic distance, which does not arise from a Riemannian metric, remains suitable for a Schwarz–Pick type of inequality even in more general settings with the existence of branch points.