Koebe function
E898494
The Koebe function is a specific univalent holomorphic function on the unit disk that extremizes several classical bounds in geometric function theory, notably serving as the extremal example in the Koebe quarter theorem.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Koebe function canonical | 6 |
| Koebe function k(z)=\frac{z}{(1-z)^2} | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991872 — 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: Koebe function Context triple: [Koebe quarter theorem, relatedTo, Koebe function]
-
A.
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.
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B.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
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C.
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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D.
Blaschke products
Blaschke products are bounded analytic functions on the unit disk formed as (finite or infinite) products of Möbius transformations that map the disk to itself, playing a central role in complex analysis and function theory.
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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: Koebe function Target entity description: The Koebe function is a specific univalent holomorphic function on the unit disk that extremizes several classical bounds in geometric function theory, notably serving as the extremal example in the Koebe quarter theorem.
-
A.
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.
-
B.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
C.
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.
-
D.
Blaschke products
Blaschke products are bounded analytic functions on the unit disk formed as (finite or infinite) products of Möbius transformations that map the disk to itself, playing a central role in complex analysis and function theory.
-
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 (44)
| Predicate | Object |
|---|---|
| instanceOf |
extremal function
ⓘ
holomorphic function ⓘ object of geometric function theory ⓘ univalent function ⓘ |
| alternativeFormula | k(z) = z + 2z^2 + 3z^3 + 4z^4 + \ ⓘ |
| attainsEqualityIn |
Koebe quarter theorem lower bound 1/4
ⓘ
distortion theorem for univalent functions ⓘ growth theorem for univalent functions ⓘ |
| belongsToClass | S (class of normalized univalent functions on unit disk) ⓘ |
| definedOn | open unit disk ⓘ |
| extremalFamily | {e^{-i\theta} k(e^{i\theta} z) : \theta \in \mathbb{R}} ⓘ |
| field |
complex analysis
ⓘ
geometric function theory ⓘ |
| hasAsymptoticBehavior |
k(-r) \to -1/4 as r \to 1^- along negative real axis
ⓘ
k(r) \to +\infty as r \to 1^- along positive real axis ⓘ |
| hasCoveringProperty | k(\mathbb{D}) \supset \{w : |w| < 1/4\} ⓘ |
| hasDerivative | k'(z) = (1 + z) / (1 - z)^3 ⓘ |
| hasFormula | k(z) = z / (1 - z)^2 ⓘ |
| hasGrowthProperty | |k(z)| \ge |z| / (1 + |z|)^2 for |z| < 1 ⓘ |
| hasMaclaurinSeries | z + 2z^2 + 3z^3 + 4z^4 + 5z^5 + \cdots ⓘ |
| hasRealBoundaryValuesOn |
(-1,0)
ⓘ
(0,1) ⓘ |
| hasRotationFamily | e^{-i\theta} k(e^{i\theta} z) ⓘ |
| hasSecondDerivative | k''(z) = 4 / (1 - z)^4 ⓘ |
| hasSingularity | pole of order 2 at z = 1 ⓘ |
| hasSlitType | slit along the negative real axis ⓘ |
| hasTaylorCoefficient | a_n = n for n \ge 1 ⓘ |
| image | \mathbb{C} \setminus (-\infty,-1/4] ⓘ |
| isExampleOf | normalized schlicht function ⓘ |
| isExtremalFor |
Bieberbach conjecture for n = 2
ⓘ
Koebe quarter theorem ⓘ covering theorems in geometric function theory ⓘ distortion estimates for univalent functions ⓘ growth estimates for univalent functions ⓘ |
| isPrototypeFor | sharp coefficient bounds in univalent function theory ⓘ |
| isUnivalentOn | {z \in \mathbb{C} : |z| < 1} ⓘ |
| maps | unit disk onto complex plane minus a slit ⓘ |
| namedAfter | Paul Koebe ⓘ |
| normalizedBy |
k'(0) = 1
ⓘ
k(0) = 0 ⓘ |
| powerSeriesExpansion | k(z) = \sum_{n=1}^{\infty} n z^n ⓘ |
| symbol | k(z) ⓘ |
| usedAs | extremal example in many univalence criteria ⓘ |
| usedToShow | sharpness of coefficient estimates for univalent functions ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: Koebe function Description of subject: The Koebe function is a specific univalent holomorphic function on the unit disk that extremizes several classical bounds in geometric function theory, notably serving as the extremal example in the Koebe quarter theorem.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Koebe function