Koebe quarter theorem
E259770
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Koebe quarter theorem canonical | 8 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364527 — 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: Koebe quarter theorem Context triple: [Riemann mapping theorem, relatedTo, Koebe quarter theorem]
-
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.
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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.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
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D.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
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E.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
- 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: Koebe quarter theorem Target entity description: 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.
-
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.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
D.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
-
E.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in geometric function theory
ⓘ
theorem in complex analysis ⓘ |
| appearsIn |
standard textbooks on complex analysis
ⓘ
standard textbooks on geometric function theory ⓘ |
| appliesTo | holomorphic injective maps on the unit disk ⓘ |
| assumes |
function is holomorphic on open unit disk
ⓘ
function is univalent on open unit disk ⓘ |
| codomainProperty | image domain contains a Euclidean disk of radius at least 1/4 ⓘ |
| concerns |
holomorphic functions
ⓘ
univalent functions ⓘ |
| concernsSet | open unit disk in the complex plane ⓘ |
| conclusion | image of unit disk contains disk of radius 1/4 centered at f(0) ⓘ |
| consequence | image of unit disk cannot be too small near f(0) ⓘ |
| domain | unit disk ⓘ |
| extremalFunction | Koebe function ⓘ |
| field |
complex analysis
ⓘ
geometric function theory ⓘ |
| givesBoundOn |
inner radius of image domain
ⓘ
size of image of univalent function ⓘ |
| hasAlternativeFormulation | for normalized univalent f with f(0)=0 and f'(0)=1, the image contains the disk of radius 1/4 centered at 0 ⓘ |
| hasLowerBound | 1/4 for radius of disk contained in image ⓘ |
| holdsFor | complex-valued functions ⓘ |
| implies |
lower bound on conformal radius at 0
ⓘ
lower bound on distance from f(0) to boundary of image domain ⓘ |
| involves | normalized univalent functions ⓘ |
| isEquivalentTo | statement about covering properties of schlicht functions ⓘ |
| mathematicalDomain | analysis ⓘ |
| mathematicalSubjectClassification | 30C45 ⓘ |
| namedAfter | Paul Koebe ⓘ |
| relatedTo |
Bieberbach conjecture
ⓘ
Koebe function ⓘ Riemann mapping theorem ⓘ distortion theorem ⓘ growth theorem for univalent functions ⓘ |
| sharpness | bound 1/4 is best possible ⓘ |
| sharpnessWitnessedBy | Koebe function ⓘ |
| states | any univalent holomorphic function on the unit disk maps it onto a domain containing a disk of radius at least one quarter ⓘ |
| topic |
conformal mapping
ⓘ
schlicht functions ⓘ |
| typeOfBound | universal bound independent of particular univalent function ⓘ |
| typicalNormalization |
f'(0)=1
ⓘ
f(0)=0 ⓘ |
| usedAs | tool in proving other results about univalent functions ⓘ |
| usedIn |
distortion theorems for univalent functions
ⓘ
geometric estimates in conformal mapping ⓘ growth estimates for univalent functions ⓘ |
| yearProved | early 20th century ⓘ |
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: Koebe quarter theorem Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.