SchwarzLemma 2002-06-05 11:33:25 2006-09-13 22:18:32 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} \newcommand{\norm}[1]{\left\|#1\right\|} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\size}[1]{| #1 |} Let $\Delta = \{z: \size{z} < 1\}$ be the open unit disk in the complex plane $\Complex$. Let $f\colon\Delta\to\Delta$ be a holomorphic function with $f(0)=0$. Then $\size{f(z)} \le \size{z}$ for all $z\in\Delta$, and $\size{f'(0)} \le 1$. If the equality $\size{f(z)}=\size{z}$ holds for \emph{any} $z\ne 0$ or $\size{f'(0)}=1$, then $f$ is a rotation: $f(z)=az$ with $\size{a}=1$. This lemma is less celebrated than the bigger guns (such as the Riemann mapping theorem, which it helps prove); however, it is one of the simplest results capturing the ``rigidity'' of holomorphic functions. No \PMlinkescapetext{similar} result exists for real functions, of course.