Maximum modulus principle

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

This fact is an application of the following pivotal fact/result/idea: open mapping theorem
View other applications of open mapping theorem OR Read a survey article on applying open mapping theorem

Statement

Suppose $U\subset \mathbb{C}$ is a domain (open connected subset). Let $f:U\to \mathbb{C}$ be a holomorphic function. The maximum modulus principle (sometimes called 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

1. Open mapping theorem

Proof

Given: $f$ is a nonconstant holomorphic function on a nonempty domain $U$.

To prove: There does not exist any $z_0\in U$ with the property that $|f(z)|\le |f(z_0)|$ for every $z\in U$>

Proof: First, by the open mapping theorem (fact (1)), $f$ is an open map.

Second, the modulus 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 $U$.