Complex Analysis/Schwarz's Lemma

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

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

• ${\displaystyle |f(z)|\leq |z|}$ for all ${\displaystyle z\in \mathbb {D} }$
• ${\displaystyle |f'(0)|\leq 1}$

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

Define ${\displaystyle g\colon \mathbb {D} \to \mathbb {C} }$ by

${\displaystyle g(z)=\left\{{\begin{array}{rr}\displaystyle {\frac {f(z)}{z}},&z\neq 0\\f'(0),&z=0\end{array}}\right.}$

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

${\displaystyle |g(z)|\leq \max _{|z|=r}|g(z)|={\frac {\max _{|z|=r}|f(z)|}{r}}\leq {\frac {1}{r}}.}$

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

• ${\displaystyle \mathrm {Aut} (\mathbb {D} )}$ using this lemma.

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