Schwarz Lemma

Schwarz Lemma 為複分析中重要定理，因數學家 Hermann Amandus Schwarz 而取名，從網路上可查詢到多種證明方式，並有多個延伸定理。最原始的定理說明如下：

Schwarz Lemma:

Let $\mathbb{D}=\{z: |z| \lt 1 \}$ be an open unit disk in the complex plane $\mathbb{C}$ and $f:\mathbb{D}\to\mathbb{D}$ be a holomorphic map such that $f(0)=0$.

Then, $|f(z)|\le z$, $\forall z\in\mathbb{D}$ and $|f'(0)|\le 1$.

Moreover, if $|f(z)|=z$ for some non-zero $z$ or $|f'(0)|=1$, then
\[ f(z)=a z\]
for some $a\in\mathbb{C}$ with $|a|=1$.

Proof. Let $g(z)=\frac{f(z)}{z}$ for any $z\neq 0$ and $g(0)=f'(0)$, then $g$ is holomorphic on $\mathbb{D}$. Now apply the Maximum Modulus Principle to $g$ on the disk $D(0,r)$ with $0\lt r\lt 1$
\[ |g(z)|\le \frac{1}{r}, 0\lt |z| \lt 1\]
Fix $z$ and let $r\to 1$, we get $|g(z)|\le 1$ which means
\[ |f(z)|\le |z|. \]