distortion theorem
E898496
The distortion theorem is a result in complex analysis that provides sharp bounds on how much a univalent (injective holomorphic) function can stretch or compress distances in the unit disk.
All labels observed (1)
| Label | Occurrences |
|---|---|
| distortion theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991892 — 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: distortion theorem Context triple: [Koebe quarter theorem, relatedTo, distortion theorem]
-
A.
Bernstein theorem
Bernstein theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
-
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.
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C.
Rouché's theorem
Rouché's theorem is a result in complex analysis that provides conditions under which two holomorphic functions have the same number of zeros inside a given contour.
-
D.
Hadamard three-circle theorem
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.
-
E.
GAGA theorems
The GAGA theorems are foundational results in algebraic geometry that rigorously relate complex algebraic varieties to their associated analytic spaces, showing an equivalence between algebraic and analytic categories under suitable conditions.
- 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: distortion theorem Target entity description: The distortion theorem is a result in complex analysis that provides sharp bounds on how much a univalent (injective holomorphic) function can stretch or compress distances in the unit disk.
-
A.
Bernstein theorem
Bernstein theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
-
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.
Rouché's theorem
Rouché's theorem is a result in complex analysis that provides conditions under which two holomorphic functions have the same number of zeros inside a given contour.
-
D.
Hadamard three-circle theorem
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.
-
E.
GAGA theorems
The GAGA theorems are foundational results in algebraic geometry that rigorously relate complex algebraic varieties to their associated analytic spaces, showing an equivalence between algebraic and analytic categories under suitable conditions.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | theorem in complex analysis ⓘ |
| alsoKnownAs | Koebe distortion theorem ⓘ |
| appliesTo |
injective holomorphic function
ⓘ
univalent function ⓘ |
| assumption |
holomorphicity on the unit disk
ⓘ
injectivity on the unit disk ⓘ normalization at the origin ⓘ |
| category | result about conformal mappings ⓘ |
| concerns |
distortion of derivatives
ⓘ
distortion of distances ⓘ distortion of moduli of values ⓘ geometric behavior of holomorphic functions ⓘ |
| domainCondition |
open unit disk
ⓘ
unit disk ⓘ |
| ensures | univalent functions cannot distort distances arbitrarily near 0 ⓘ |
| equalityCase | rotations of the Koebe function ⓘ |
| extremalFunction | Koebe function ⓘ |
| extremalFunctionExample | k(z)=z/(1-z)^2 ⓘ |
| field | complex analysis ⓘ |
| generalizationOf | basic derivative estimates for bounded holomorphic functions ⓘ |
| gives |
sharp bounds on compression
ⓘ
sharp bounds on stretching ⓘ two-sided estimates for |f'(z)| ⓘ two-sided estimates for |f(z)| ⓘ |
| hasVariant |
distortion theorem for schlicht functions
ⓘ
distortion theorem in the class S of normalized univalent functions ⓘ |
| implies |
control of derivative growth of univalent functions
ⓘ
control of image size of univalent functions ⓘ local quasi-isometry properties near the origin ⓘ |
| involves | radial parameter r=|z| ⓘ |
| isSharp | yes ⓘ |
| mathematicalSubjectClassification | 30C45 ⓘ |
| normalizationCondition |
f'(0)=1
ⓘ
f(0)=0 ⓘ |
| provides | bounds depending only on |z| ⓘ |
| relatedTo |
Bieberbach conjecture
ⓘ
Koebe quarter theorem ⓘ growth theorem ⓘ |
| type | metric distortion estimate ⓘ |
| typicalForm |
If f is univalent on the unit disk with f(0)=0 and f'(0)=1, then for |z|=r<1, (1-r)^3 ≤ |f'(z)| ≤ (1+r)^3/(1-r)^3
ⓘ
If f is univalent on the unit disk with f(0)=0 and f'(0)=1, then r/(1+r)^2 ≤ |f(z)| ≤ r/(1-r)^2 for |z|=r<1 ⓘ |
| typicalNormalizationClass | S = {f univalent on unit disk : f(0)=0, f'(0)=1} ⓘ |
| usedFor |
bounding coefficients of univalent functions
ⓘ
studying boundary behavior of conformal maps ⓘ |
| usedIn |
geometric function theory
ⓘ
proofs of growth and covering theorems ⓘ theory of univalent functions ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: distortion theorem Description of subject: The distortion theorem is a result in complex analysis that provides sharp bounds on how much a univalent (injective holomorphic) function can stretch or compress distances in the unit disk.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.