# Schwarz Lemma & Maximum Modulus Principle

## Schwarz Lemma

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

Then, $|f(z)|\leq |z|$ for all $z\in D$ and $|f'(0)\leq 1$.

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

## Maximum modulus principle

Let $f$ be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. If $z_0$ is a point in $D$ such that $|f(z_0)|\geq|f(z)|$ for all $z$ in a neighborhood of $z_0$, then the function $f$ is constant on $D$.

Informally, the modulus $|f|$ cannot exhibit a true local maximum that is properly within the domain of $f$.

## Author: mathtuition88

http://mathtuition88.com

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