Complex Analysis/Schwarz's Lemma

The Schwarz Lemma is a statement about the growth behavior of holomorphic functions on the unit disk.

Statement

Let $𝔻:=\{z\in ℂ:|z|<1\}$ be the unit disk, and let $f:𝔻\to 𝔻$ be holomorphic with $f(0)=0$. Then the following hold:

• $|f(z)|\le |z|$ for all $z\in 𝔻$
• $|f^{\prime }(0)|\le 1$

• If $|f^{\prime }(0)|=1$ or $|f(z_0)|=|z_0|$ for some $z_0\in 𝔻\setminus 0$, then $f$ is a rotation, i.e., there exists a $\lambda$ with $|\lambda |=1$ such that $f(z)=\lambda z$ for all $z\in 𝔻$.

Proof

Define $g:𝔻\to ℂ$ by

$g(z)=\{\begin{array}{cc}\frac{f(z)}{z},& z\ne 0\\ f^{\prime }(0),& z=0\end{array}$

Then $g$ is continuous and therefore, by the Riemann Removability Theorem, also holomorphic. Let $r<1$. By the Maximum Principle, for $|z|\le r$, we have:

$|g(z)|\le \underset{|z|=r}{\max }|g(z)|=\frac{\underset{|z|=r}{\max }|f(z)|}{r}\le \frac{1}{r}.$

As $r\to 1$, it follows that $|g(z)|\le 1$, hence $|f(z)|\le |z|$ for all $z\in 𝔻$, proving the first two statements. If equality holds in either case, then $|g|$ has a local maximum in the interior of $𝔻$. By the Maximum Modulus Principle, $g$ must be constant. This constant $\lambda$ has modulus $1$, and the claim follows. See Fischer, p. 286.

See Also

• $\mathrm{A}\mathrm{u}\mathrm{t}(𝔻)$ using this lemma.

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Lemma von Schwarz - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Lemma_von_Schwarz

• Date: 01/08/2024

de:Kurs:Funktionentheorie/Lemma von Schwarz