High Quality Content by WIKIPEDIA articles! In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincare half-plane model. It states that the Poincare metric is distance-decreasing on harmonic functions. The theorem states that every holomorphic function on the unit disk U, or the upper half-plane H, with distances defined by the Poincare 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 functions f:Urightarrow U, one has rho(f(z_1),f(z_2)) leq rho(z_1,z_2) for points z_1,z_2 in U and Poincare distance ?. For any tangent vector T, the hyperbolic length of the tangent vector does not increase: |f^*(T)| leq |T|.

Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.