Schwarz–Ahlfors–Pick theorem

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In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. It states that the Poincaré metric is distance-decreasing on harmonic functions.

The theorem states that every holomorphic automorphism of the unit disk U, or the upper half-plane H, with distances defined by the Poincaré metric, is a contraction mapping. That is, every such analytic mapping will not increase the distance between points. Stated more precisely:

Theorem: (Schwarz–Ahlfors–Pick) For all holomorphic automorphisms , one has for points and Poincaré distance ρ.

For any tangent vector T, the hyperbolic length of the tangent vector does not increase: