disc . Then, the map is a holomorphic map from onto . Using the Schwarz–Pick theorem on this map, and finally simplifying the results by using the formula for , we get the desired result. Also, for all , If equality holds for either the one or the other expressions, then must be a Möbius transformation with real coefficients. That is, if equality holds, then with and . Proof of Schwarz–Pick theorem The proof of the Schwarz–Pick theorem follows from Schwarz's lemma and the fact that a Möbius transformation of the form maps the unit circle to itself. Fix and define the Möbius transformations Since and the Möbius transformation is invertible, the composition maps to and the unit disk is mapped into itself. Thus we can apply Schwarz's lemma, which is to say Now calling (which will still be in the unit disk) yields the desired conclusion To prove the second part of the theorem, we rearrange the left-hand side into the difference quotient and let tend to . Further generalizations and related results The Schwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds. De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of at in case is injective; that is, univalent. The Koebe 1/4 theorem provides a related estimate in the case that is univalent. See also Nevanlinna–Pick interpolation References Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. (See Section 2.3) Riemann surfaces Lemmas in analysis Theorems in complex analysis Articles containing proofs