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Minimum Modulus Principle
Definition: Let f be analytic on a domain U subset= C, and assume that f never vanishes. Then if there is a point z_0 in U such that |f(z_0)|<=|f(z)| for all z in U, then f is constant. Let U subset= 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 |f| on U^_ (which always exists) must occur on partialU. In other words, min_(U^_)|f|=min_(partialU)|f|.