Maximum modulus principle

Statement

Suppose $U\subset \mathbb{C}$ is a domain (open connected subset). Let $f:U\to \mathbb{C}$ be a holomorphic function. The maximum principle states that if there exists a $z_0\in U$, such that for all $z\in U$, we have:

$|f(z)|\le |f(z_0)|$

Then, $f$ is a constant function.

Facts used

• Open mapping theorem

Proof

Suppose $f$ is a nonconstant holomorphic function on a nonempty domain $U$. We'll show that $f$ cannot have a maximum.

First, by the open mapping theorem, $f$ is an open map.

Also, observe that the map $|\cdot |:\mathbb{C}\to [0,\mathrm{\infty })$ is an open map. Thus, the composite map $\left|f\right|:U\to [0,\mathrm{\infty })$ given by $z\mapsto |f(z)|$ is also an open map. Thus, under this map, the image of $U$ must be an open connected subset of $[0,\mathrm{\infty })$ so it must be of the form $[0,a)$ where $a\in \mathbb{R}$ or $a=\mathrm{\infty }$. Hence, there cannot be a maximum within $U$.