Maximum modulus principle

In mathematics, the maximum modulus principle in complex analysis states that if ${\displaystyle f}$ is a holomorphic function, then the modulus ${\displaystyle |f|}$ cannot exhibit a strict maximum that is strictly within the domain of ${\displaystyle f}$.

In other words, either ${\displaystyle f}$ is locally a constant function, or, for any point ${\displaystyle z_{0}}$ inside the domain of ${\displaystyle f}$ there exist other points arbitrarily close to ${\displaystyle z_{0}}$ at which ${\displaystyle |f|}$ takes larger values.