In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions. Statement Let be the open unit disk in the complex plane centered at the origin, and let be a holomorphic map such that and on . Then for all , and . Moreover, if for some non-zero or , then for some with . Proof The proof is a straightforward application of the maximum modulus principle on the function which is holomorphic on the whole of , including at the origin (because is differentiable at the origin and fixes zero). Now if denotes the closed disk of radius centered at the origin, then the maximum modulus principle implies that, for , given any , there exists on the boundary of such that As we get . Moreover, suppose that for some non-zero , or . Then, at some point of . So by the maximum modulus principle, is equal to a constant such that . Therefore, , as desired. Schwarz–Pick theorem A variant of the Schwarz lemma, known as the Schwarz–Pick theorem (after Georg Pick), characterizes the analytic automorphisms of the unit disc, i.e. bijective holomorphic mappings of the unit disc to itself: Let be holomorphic. Then, for all , and, for all , The expression is the distance of the points , in the Poincaré metric, i.e. the metric in the Poincaré disc model for hyperbolic geometry in dimension two. The Schwarz–Pick theorem then essentially states that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric. If equality holds throughout in one of the two inequalities above (which is equivalent to saying that the holomorphic map preserves the distance in the Poincaré metric), then must be an analytic automorphism of the unit disc, given by a Möbius transformation mapping the unit disc to itself. An analogous statement on the upper half-plane can be made as follows: Let be holomorphic. Then, for all , This is an easy consequence of the Schwarz–Pick theorem mentioned above: One just needs to remember that the Cayley transform maps the upper half-plane conformally onto the unit