Let f be analytic on a domain U⊆C, and assume that f never vanishes. 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. Let U⊆C be a bounded domain, let f be a continuous function on the closed set U^_ that is analytic on U, and assume that f never vanishes on U^_. Then the minimum value of left bracketing bar f right bracketing bar on U^_ (which always exists) must occur on dU. In other words, min_(U^_) left bracketing bar f right bracketing bar = min_(dU) left bracketing bar f right bracketing bar .

complex modulus | maximum modulus principle