Let be a domain, and let be an analytic function on . Then if there is a point such that for all , then is constant. The following slightly sharper version can also be formulated. Let be a domain, and let be an analytic function on . Then if there is a point at which has a local maximum, then is constant.

Furthermore, let be a bounded domain, and let be a continuous function on the closed set that is analytic on . Then the maximum value of on (which always exists) occurs on the boundary . In other words,

The maximum modulus theorem is not always true on an unbounded domain.