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:
“Most personal correspondence of today consists of letters the first half of which are given over to an indexed statement of why the writer hasn’t written before, followed by one paragraph of small talk, with the remainder devoted to reasons why it is imperative that the letter be brought to a close.”
—Robert Benchley (1889–1945)
“Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.”
—Jean Ross Peterson (20th century)
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)