Hadamard three-circle theorem

E334042

result in complex function theory theorem in complex analysis

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.

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All labels observed (2)

Label	Occurrences
Hadamard three-circle theorem canonical	2
Hadamard three-lines theorem	2

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]

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
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-lines theorem ⓘ linked to: Hadamard three-circle theorem Phragmén–Lindelöf principle ⓘ linked to: Lindelöf theorem in complex analysis 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

Referenced by (4)

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

Jacques Hadamard → knownFor → Hadamard three-circle theorem ⓘ

Hadamard three-circle theorem → relatedTo → Hadamard three-lines theorem ⓘ

linked to: Hadamard three-circle theorem

Inequalities for analytic functions → relatedTo → Hadamard three-circle theorem ⓘ

Riesz–Thorin interpolation theorem → uses → Hadamard three-lines theorem ⓘ