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 <math>f:U\rightarrow U</math>, one has <math>\rho(f(z_1),f(z_2)) \leq \rho(z_1,z_2)</math> for points <math>z_1,z_2 \in U</math> and Poincaré distance <math>\rho.</math>
For any tangent vector T, the hyperbolic length of the tangent vector does not increase: <math>|f^*(T)| \leq |T|.</math>
