Bloch theorem
E898493
Bloch theorem is a fundamental result in complex analysis stating that any holomorphic function on the unit disk with bounded derivative maps some subdisk onto a disk of a universal minimum radius, known as Bloch's constant.
Statements (36)
| Predicate | Object |
|---|---|
| instanceOf | theorem in complex analysis ⓘ |
| appliesTo | holomorphic functions on simply connected domains via conformal maps ⓘ |
| assumption |
derivative of the function is bounded on the unit disk
ⓘ
function is holomorphic on the unit disk ⓘ |
| concerns | holomorphic functions ⓘ |
| conclusion | there exists a subdisk of the unit disk whose image contains a disk of radius at least Bloch constant ⓘ |
| domainCondition | unit disk in the complex plane ⓘ |
| field | complex analysis ⓘ |
| guarantees | existence of a universal positive lower bound for the radius of an image disk ⓘ |
| hasUnknownParameter | exact value of Bloch constant is not known ⓘ |
| historicalPeriod | early 20th century mathematics ⓘ |
| implies | normal families type estimates for holomorphic functions ⓘ |
| introduces | Bloch constant NERFINISHED ⓘ |
| involves |
derivative estimates for holomorphic functions
ⓘ
image disks under holomorphic maps ⓘ subdisks of the unit disk ⓘ |
| language | mathematical analysis ⓘ |
| lowerBound | Bloch constant has known positive lower bounds NERFINISHED ⓘ |
| mathematicalSubjectClassification |
30C80
ⓘ
30D45 ⓘ |
| namedAfter | André Bloch NERFINISHED ⓘ |
| objectOfStudy | local behavior of holomorphic functions ⓘ |
| quantifier |
existential subdisk of the unit disk
ⓘ
universal constant independent of the particular holomorphic function ⓘ |
| relatedTo |
Bloch constant
NERFINISHED
ⓘ
Koebe quarter theorem NERFINISHED ⓘ Landau theorem NERFINISHED ⓘ Schwarz lemma NERFINISHED ⓘ |
| statementForm | for every holomorphic function with bounded derivative on the unit disk there exists a point and radius such that the image contains a disk of universal radius ⓘ |
| typeOf |
distortion theorem
ⓘ
existence theorem ⓘ |
| upperBound | Bloch constant has known finite upper bounds NERFINISHED ⓘ |
| usedIn |
conformal mapping theory
ⓘ
geometric function theory ⓘ theory of normal families ⓘ value distribution theory ⓘ |
Referenced by (1)
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