Hadamard three-circle theorem
E334042
The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hadamard three-circle theorem canonical | 1 |
| Hadamard three-lines theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3167252 — 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: Hadamard three-circle theorem Context triple: [Jacques Hadamard, knownFor, Hadamard three-circle theorem]
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A.
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.
-
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.
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.
-
D.
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.
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E.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
- 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: Hadamard three-circle theorem Target entity description: The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
-
A.
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.
-
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.
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.
-
D.
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.
-
E.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in complex function theory
ⓘ
theorem in complex analysis ⓘ |
| appearsIn | standard graduate texts on complex analysis ⓘ |
| appliesTo |
analytic functions
ⓘ
holomorphic functions ⓘ |
| assumes |
function is holomorphic on an annulus
ⓘ
function is holomorphic on and between three concentric circles ⓘ |
| category | theorems about analytic function growth ⓘ |
| compares | maximum modulus on three circles of different radii ⓘ |
| contrastWith |
Cauchy estimates
ⓘ
Liouville's theorem ⓘ |
| describes | behavior of maximum modulus between three concentric circles ⓘ |
| domain | complex plane ⓘ |
| field | complex analysis ⓘ |
| generalizationOf | properties of subharmonic functions on annuli ⓘ |
| hasVariant | n-dimensional analogues for harmonic and subharmonic functions ⓘ |
| holdsFor |
entire functions
ⓘ
holomorphic functions on annuli ⓘ |
| implies |
growth of holomorphic functions is controlled between circles
ⓘ
log M(r) lies below line segment joining (log r1, log M(r1)) and (log r2, log M(r2)) ⓘ maximum modulus on intermediate circle is bounded by values on inner and outer circles ⓘ |
| involves |
concentric circles
ⓘ
logarithmic convexity ⓘ maximum modulus ⓘ |
| logicalForm | convexity inequality for a function of log radius ⓘ |
| mathematicsSubjectClassification | 30A10 ⓘ |
| namedAfter | Jacques Hadamard ⓘ |
| notation | M(r) denotes maximum of |f(z)| on |z|=r ⓘ |
| prerequisite |
Cauchy integral formula
ⓘ
maximum modulus principle ⓘ |
| relatedConcept |
convex function
ⓘ
entire function growth order ⓘ subharmonic function ⓘ |
| relatedTo |
Hadamard three-circle theorem
self-linksurface differs
ⓘ
surface form:
Hadamard three-lines theorem
Lindelöf theorem in complex analysis ⓘ
surface form:
Phragmén–Lindelöf principle
maximum modulus principle ⓘ |
| states | logarithm of the maximum modulus is a convex function of the logarithm of the radius ⓘ |
| timePeriod | early 20th century ⓘ |
| type | inequality theorem ⓘ |
| typicalFormulation | for 0<r1<r<r2, log M(r) is convex in log r ⓘ |
| usedFor |
classification of entire functions by order and type
ⓘ
growth estimates of entire functions ⓘ uniqueness results in complex analysis ⓘ |
| usedIn |
analytic continuation arguments
ⓘ
complex potential theory ⓘ value distribution theory ⓘ |
How these facts were elicited
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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: Hadamard three-circle theorem Description of subject: The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Hadamard three-circle theorem
→
relatedTo
→
Hadamard three-circle theorem
self-linksurface differs
ⓘ
this entity surface form:
Hadamard three-lines theorem