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:

$|f(z)|\leq |f(z_{0})|$

Then, $f$ is a constant function.

Facts used

• Open mapping theorem