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.
Read more about this topic: Schwarz Lemma
Famous quotes containing the word statement:
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“After the first powerful plain manifesto
The black statement of pistons, without more fuss
But gliding like a queen, she leaves the station.”
—Stephen Spender (1909–1995)
“Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
—Charles Sanders Peirce (1839–1914)