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