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 will comprise the skepticisms, as well as the faiths of society, and out of unbeliefs a creed shall be formed. For, skepticisms are not gratuitous or lawless, but are limitations of the affirmative statement, and the new philosophy must take them in, and make affirmations outside of them, just as much as must include the oldest beliefs.”
—Ralph Waldo Emerson (1803–1882)
“The parent is the strongest statement that the child hears regarding what it means to be alive and real. More than what we say or do, the way we are expresses what we think it means to be alive. So the articulate parent is less a telling than a listening individual.”
—Polly Berrien Berends (20th century)
“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)