Schwarz Lemma - Statement

Statement

Schwarz Lemma. Let D = {z : |z| < 1} be the open unit disk in the complex plane C centered at the origin and let f : D → D be a holomorphic map such that f(0) = 0. Then, |f(z)| ≤ |z| for all z in D and |f′(0)| ≤ 1. Moreover, if |f(z)| = |z| for some non-zero z or |f′(0)| = 1, then f(z) = az for some a in C with |a| = 1.

Note. Some authors replace the condition f : D → D with |f(z)| ≤ 1 for all z in D (where f is still holomorphic in D). The two versions can be shown to be equivalent through an application of the maximum modulus principle.