Schwarz–Pick theorem
E899966
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Schwarz–Ahlfors lemma | 1 |
| Schwarz–Pick theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991824 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Schwarz–Pick theorem Context triple: [Schwarz lemma, generalization, Schwarz–Pick theorem]
-
A.
Schwarz lemma
Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
-
B.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
C.
Bieberbach conjecture
The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
-
D.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
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E.
Koebe quarter theorem
The Koebe quarter theorem is a fundamental result in complex analysis stating that any univalent holomorphic function on the unit disk maps it onto a domain containing a disk of radius one quarter, providing a sharp bound on the size of the image.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Schwarz–Pick theorem Target entity description: 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.
-
A.
Schwarz lemma
Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
-
B.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
C.
Bieberbach conjecture
The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
-
D.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
E.
Koebe quarter theorem
The Koebe quarter theorem is a fundamental result in complex analysis stating that any univalent holomorphic function on the unit disk maps it onto a domain containing a disk of radius one quarter, providing a sharp bound on the size of the image.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Schwarz–Pick theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Schwarz–Ahlfors lemma