---
tag: 048F2P90WR
kind: statement
role: theorem
name: Maximum modulus principle
aliases: [maximum-modulus-theorem]
msc: [30C80]
uses: [0DG0Z72CQD, 097KNJFJ7V, 0HYPR64JM2, 0YBVKMYX0W]
sources:
  - url: https://en.wikipedia.org/wiki/Maximum_modulus_principle
    locator: Formal statement
  - url: https://en.wikipedia.org/wiki/Maximum_modulus_principle
    locator: Using Gauss's mean value theorem
same_as:
  wikidata: Q1050230
  wikipedia: Maximum modulus principle
---

## Statement

Let $D \subseteq \mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Assume that $|f|$ has a local maximum at a point $z_0 \in D$: there is $r > 0$ such that $|f(z)| \le |f(z_0)|$ for each $z$ with $|z - z_0| < r$.
Then $f$ is constant on $D$.

## Remarks

Proof idea: the mean value formula $f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + \rho e^{i\theta}) \, d\theta$ for small $\rho$ forces $|f| = |f(z_0)|$ near $z_0$. Then $f$ is constant near $z_0$, and on $D$ by the [[identity theorem|0YBVKMYX0W]].
Corollary: if $D$ is bounded and $f$ is continuous on the closure $\overline{D}$ and holomorphic on $D$, then $\max_{\overline{D}} |f|$ is attained at a point of the boundary of $D$.
The minimum of $|f|$ can be inside $D$, but only at a zero of $f$. For example, $|z|$ has its minimum on the unit disc at $0$.

## Explanations

### L1 human

The size $|f|$ of a nonconstant holomorphic function has no highest point inside its domain.
For example, on the closed unit disc, $|z^2 + 1|$ is largest on the boundary circle, at $z = \pm 1$, and it equals $2$ there.