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:
“The new statement is always hated by the old, and, to those dwelling in the old, comes like an abyss of skepticism.”
—Ralph Waldo Emerson (1803–1882)
“One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
—Rebecca West (1892–1983)
“Truth is used to vitalize a statement rather than devitalize it. Truth implies more than a simple statement of fact. “I don’t have any whisky,” may be a fact but it is not a truth.”
—William Burroughs (b. 1914)