Schwarz lemma
E259769
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.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Schwarz lemma canonical | 5 |
| Schwarz lemma without the condition f(0) = 0 via Möbius transformations | 1 |
| Schwarz-type lemma | 1 |
| Schwarz–Pick theorem | 1 |
| Schwarz’s lemma | 1 |
| boundary Schwarz lemma | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364526 — 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 lemma Context triple: [Riemann mapping theorem, relatedTo, Schwarz lemma]
-
A.
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.
-
B.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
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C.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
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D.
Cauchy integral formula
The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
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E.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
- 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 lemma Target entity description: 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.
-
A.
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.
-
B.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
C.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
D.
Cauchy integral formula
The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
-
E.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in complex analysis ⓘ |
| alsoKnownAs |
Schwarz’s lemma
ⓘ
linked to:
Schwarz lemma
|
| appliesTo |
holomorphic functions
ⓘ
holomorphic self-maps of the unit disk ⓘ |
| assumption |
f is analytic on the open unit disk
ⓘ
|f(z)| ≤ 1 for all z in the unit disk ⓘ |
| category | theorem about bounded analytic functions ⓘ |
| characterizes | holomorphic automorphisms of the unit disk fixing the origin ⓘ |
| codomainCondition | function maps the unit disk into itself ⓘ |
| conclusion |
|f'(0)| ≤ 1
ⓘ
|f(z)| ≤ |z| for all z in the unit disk ⓘ |
| consequence |
derivative at the origin of a self-map of the disk fixing 0 is bounded by 1 in modulus
ⓘ
origin is a fixed point of extremal maps ⓘ |
| coreInequality | |f'(0)| ≤ 1 and |f(z)| ≤ |z| ⓘ |
| domainCondition | function holomorphic on the open unit disk ⓘ |
| equalityCaseDescription | f(z) = e^{iθ} z for some real θ ⓘ |
| equalityCondition |
if |f'(0)| = 1 then f is a rotation
ⓘ
if |f(z)| = |z| for some nonzero z then f is a rotation ⓘ |
| field | complex analysis ⓘ |
| generalization |
Schwarz–Ahlfors lemma
ⓘ
linked to:
Schwarz–Pick theorem
Schwarz–Pick theorem ⓘ |
| hasVariant |
Schwarz lemma without the condition f(0) = 0 via Möbius transformations
ⓘ
linked to:
Schwarz lemma
boundary Schwarz lemma ⓘ
linked to:
Schwarz lemma
|
| historicalPeriod | late 19th century mathematics ⓘ |
| holdsIn | unit disk in the complex plane ⓘ |
| implies | maximum modulus principle in special cases ⓘ |
| importance | fundamental tool in geometric function theory ⓘ |
| involvesObject |
complex derivative at the origin
ⓘ
open unit disk {z ∈ ℂ : |z| < 1} ⓘ |
| languageOfName | German ⓘ |
| mathematicalSubjectClassification |
MSC 30C80
ⓘ
MSC 30D05 ⓘ |
| namedAfter |
Hermann Schwarz
ⓘ
linked to:
Hermann Amandus Schwarz
|
| normalizationCondition | f(0) = 0 ⓘ |
| relatedConcept |
conformal self-maps of the unit disk
ⓘ
hyperbolic metric on the unit disk ⓘ |
| relatedResult |
Bloch theorem
ⓘ
Koebe quarter theorem ⓘ Little Picard theorem ⓘ
linked to:
Picard theorem
|
| typicalProofMethod |
application of the maximum modulus principle
ⓘ
consideration of auxiliary function f(z)/z ⓘ |
| usedFor |
bounding derivatives of bounded holomorphic functions
ⓘ
proving rigidity results for holomorphic maps ⓘ studying fixed points of holomorphic self-maps ⓘ |
| usedInProofOf |
Riemann mapping theorem
ⓘ
Schwarz–Pick theorem ⓘ
linked to:
Schwarz lemma
|
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Schwarz lemma Description of subject: 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.
Referenced by (10)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Schwarz lemma
Schwarz lemma
→
hasVariant
→
Schwarz lemma without the condition f(0) = 0 via Möbius transformations
ⓘ
linked to: Schwarz lemma
linked to: Schwarz lemma
linked to: Schwarz lemma
subject linked to:
Functions of One Complex Variable by John B. Conway
linked to: Schwarz lemma