Let U⊆C be a domain, and let f be an analytic function on U. Then if there is a point z_0 element U such that left bracketing bar f(z_0) right bracketing bar >= left bracketing bar f(z) right bracketing bar for all z element U, then f is constant. The following slightly sharper version can also be formulated. Let U⊆C be a domain, and let f be an analytic function on U. Then if there is a point z_0 element U at which left bracketing bar f right bracketing bar has a local maximum, then f is constant. Furthermore, let U⊆C be a bounded domain, and let f be a continuous function on the closed set U^_ that is analytic on U. Then the maximum value of left bracketing bar f right bracketing bar on U^_ (which always exists) occurs on the boundary dU. In other words, max_(U^_) left bracketing bar f right bracketing bar = max_(dU) left bracketing bar f right bracketing bar . The maximum modulus theorem is not always true on an unbounded domain.

complex modulus | minimum modulus principle