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.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | theorem in complex analysis ⓘ |
| alsoKnownAs | Koebe distortion theorem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Koebe quarter theorem NERFINISHED ⓘ 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} GENERATED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.