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 $f:U\backslash rightarrow\; U$, one has $\backslash rho(f(z\_1),f(z\_2))\; \backslash leq\; \backslash rho(z\_1,z\_2)$ for points $z\_1,z\_2\; \backslash in\; U$ and Poincaré distance $\backslash rho.$

For any tangent vector T, the hyperbolic length of the tangent vector does not increase: $|f^*(T)|\; \backslash leq\; |T|.$

Hyperbolic geometry | Differential geometry | Riemann surfaces | Mathematical theorems