Schwarz lemma

Let Δ={z:|z|<1} be the open unit disk in the complex planeMathworldPlanetmath . Let f:ΔΔ be a holomorphic functionMathworldPlanetmath with f(0)=0. Then |f(z)||z| for all zΔ, and |f(0)|1. If the equality |f(z)|=|z| holds for any z0 or |f(0)|=1, then f is a rotation: f(z)=az with |a|=1.

This lemma is less celebrated than the bigger guns (such as the Riemann mapping theoremMathworldPlanetmath, which it helps prove); however, it is one of the simplest results capturing the “rigidity” of holomorphic functions. No result exists for real functions, of course.

Title Schwarz lemmaMathworldPlanetmath
Canonical name SchwarzLemma
Date of creation 2013-03-22 12:44:37
Last modified on 2013-03-22 12:44:37
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Theorem
Classification msc 30C80