Maximum modulus principle

"Maximal principle" redirects here; not to be confused with Hausdorff maximal principle.

A plot of the modulus of $\cos(z)$ (in red) for $z$ in the unit disk centered at the origin (shown in blue). As predicted by the theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge).

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

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