Maximum Modulus Principle

A theorem in complex analysis that asserts: if f is a nonconstant holomorphic function on a connected open set, then the absolute value |f| cannot attain a local (or interior) maximum; any global maximum of |f| on a bounded domain occurs on the domain's boundary unless f is constant.

A principle in complex analysis which states that a nonconstant holomorphic function on a connected open domain cannot attain a strict local maximum of its modulus in the interior; if the modulus attains a global maximum on the domain, the function is constant. Equivalently, |f| is subharmonic and achieves its maximum on the boundary.