Complex Analysis/Schwarz's Lemma

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

Statement

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

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

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

Proof

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

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

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

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

As $r\to 1$, it follows that $|g(z)|\leq 1$, hence $|f(z)|\leq |z|$ for all $z\in \mathbb {D}$, proving the first two statements. If equality holds in either case, then $|g|$ has a local maximum in the interior of $\mathbb {D}$. 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

• $\mathrm {Aut} (\mathbb {D} )$ 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