Let be the open unit disk in the complex plane . Let be a holomorphic function with . Then for all , and . If the equality holds for any or , then is a rotation: with .

This lemma is less celebrated than the bigger guns (such as the Riemann mapping theorem, which it helps prove); however, it is one of the simplest results capturing the “rigidity” of holomorphic functions. No similar result exists for real functions, of course.