Schwarz–Pick theorem

E899966

theorem in complex analysis

The Schwarz–Pick theorem is a fundamental result in complex analysis that characterizes holomorphic self-maps of the unit disk by showing they are distance-decreasing with respect to the hyperbolic (Poincaré) metric.

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All labels observed (2)

Label	Occurrences
Schwarz–Ahlfors lemma	1
Schwarz–Pick theorem canonical	1

Statements (47)

Predicate	Object
instanceOf	theorem in complex analysis ⓘ
appliesTo	holomorphic map f from unit disk to unit disk ⓘ
assumes	holomorphicity on the open unit disk ⓘ map takes values in the closed unit disk ⓘ
category	results about holomorphic self-maps ⓘ
characterizes	distance-decreasing property of holomorphic self-maps of the unit disk ⓘ
codomain	unit disk ⓘ
concerns	holomorphic functions ⓘ holomorphic self-maps of the unit disk ⓘ
conclusion	map is a strict contraction in the hyperbolic metric unless it is an automorphism ⓘ
domain	unit disk ⓘ
field	complex analysis ⓘ
formalizes	hyperbolic non-expansiveness of holomorphic maps ⓘ
generalizes	Schwarz lemma NERFINISHED ⓘ
hasConsequence	rigidity of automorphisms of the unit disk ⓘ uniqueness of holomorphic self-maps with prescribed values and derivatives at points ⓘ
hasEqualityCondition	equality holds if and only if f is a Möbius automorphism of the unit disk ⓘ
hasVersion	global form involving hyperbolic distance ⓘ infinitesimal form involving derivatives ⓘ
historicalOrigin	work of Georg Pick in the early 20th century ⓘ work of Hermann Schwarz in the 19th century ⓘ
holdsIn	unit disk model of the hyperbolic plane ⓘ
implies	boundary behavior constraints for holomorphic self-maps of the unit disk ⓘ holomorphic self-maps of the unit disk are 1-Lipschitz for the hyperbolic metric ⓘ holomorphic self-maps of the unit disk are contractions for the hyperbolic metric ⓘ holomorphic self-maps of the unit disk are non-expansive with respect to the Poincaré metric ⓘ holomorphic self-maps of the unit disk strictly decrease hyperbolic distance unless they are automorphisms ⓘ
inspired	general Schwarz–Pick lemmas on complex manifolds ⓘ
involves	Möbius transformations NERFINISHED ⓘ automorphisms of the unit disk ⓘ
isToolFor	proving normal family results ⓘ studying fixed points of holomorphic self-maps of the disk ⓘ
namedAfter	Georg Pick NERFINISHED ⓘ Hermann Schwarz NERFINISHED ⓘ
relatedTo	Carathéodory metric NERFINISHED ⓘ Kobayashi metric NERFINISHED ⓘ Nevanlinna–Pick interpolation NERFINISHED ⓘ Riemann mapping theorem NERFINISHED ⓘ
statesInequality	hyperbolic distance between f(z1) and f(z2) is at most hyperbolic distance between z1 and z2 ⓘ \|f'(z)\| ≤ (1 - \|f(z)\|^2) / (1 - \|z\|^2) for z in the unit disk ⓘ
typeOf	Schwarz-type lemma NERFINISHED ⓘ
usedIn	Teichmüller theory NERFINISHED ⓘ geometric function theory ⓘ hyperbolic geometry of Riemann surfaces ⓘ iteration theory of holomorphic maps ⓘ
usesMetric	Poincaré metric NERFINISHED ⓘ hyperbolic metric on the unit disk ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Schwarz lemma → generalization → Schwarz–Pick theorem ⓘ

this entity surface form: Schwarz–Ahlfors lemma