Bieberbach conjecture
E898495
The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Bieberbach conjecture canonical | 2 |
| de Branges theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991891 — 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: Bieberbach conjecture Context triple: [Koebe quarter theorem, relatedTo, Bieberbach conjecture]
-
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.
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.
-
C.
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.
-
D.
Lindelöf hypothesis
The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
-
E.
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.
- 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: Bieberbach conjecture Target entity description: The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
-
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.
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.
-
C.
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.
-
D.
Lindelöf hypothesis
The Lindelöf hypothesis is an unproven conjecture in analytic number theory about the growth rate of the Riemann zeta function along the critical line, with deep implications for the distribution of prime numbers.
-
E.
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.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
mathematical theorem ⓘ result in complex analysis ⓘ |
| alsoKnownAs |
de Branges theorem
ⓘ
linked to:
Bieberbach conjecture
|
| assumes |
f is holomorphic on the open unit disk
ⓘ
f is injective on the open unit disk ⓘ f'(0)=1 ⓘ f(0)=0 ⓘ |
| author | Ludwig Bieberbach ⓘ |
| concerns |
bounds on Taylor coefficients
ⓘ
normalized univalent holomorphic functions on the unit disk ⓘ |
| conclusion | For normalized univalent functions on the unit disk, the nth Taylor coefficient has modulus at most n ⓘ |
| countryOfOrigin | Germany ⓘ |
| domain | unit disk ⓘ |
| equalityCase |
Koebe function k(z)=\frac{z}{(1-z)^2}
ⓘ
linked to:
Koebe function
functions obtained from the Koebe function by rotation ⓘ |
| extremalFunction |
Koebe function
ⓘ
rotations of the Koebe function ⓘ |
| field |
complex analysis
ⓘ
geometric function theory ⓘ |
| implies |
area theorems for univalent functions
ⓘ
distortion theorems for univalent functions ⓘ growth estimates for univalent functions ⓘ |
| influenced | development of geometric function theory in the 20th century ⓘ |
| involves |
analytic functions normalized at the origin
ⓘ
coefficient inequalities ⓘ extremal problems in conformal mapping ⓘ |
| mainSubject |
Taylor coefficients
ⓘ
univalent functions ⓘ |
| methodOfProof |
Hilbert space of entire functions
ⓘ
Loewner chain method ⓘ |
| namedAfter | Ludwig Bieberbach ⓘ |
| partialResultsBy |
Aurel Wintner
ⓘ
Charles Loewner ⓘ J. A. Jenkins ⓘ Ludwig Bieberbach ⓘ Menahem Schiffer ⓘ Paul Koebe ⓘ Y. Komatu ⓘ Zeev Nehari ⓘ |
| provedBy | Louis de Branges ⓘ |
| relatedTo |
Koebe quarter theorem
ⓘ
Loewner differential equation ⓘ Schlicht functions ⓘ area theorem ⓘ univalent function theory ⓘ |
| sharpness | The bound |a_n| \le n is best possible ⓘ |
| statement | If f(z)=z+\sum_{n=2}^{\infty} a_n z^n is univalent on the unit disk, then |a_n| \le n for all n \ge 2 ⓘ |
| status | proved ⓘ |
| yearProposed | 1916 ⓘ |
| yearProved | 1984 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Bieberbach conjecture Description of subject: The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Bieberbach conjecture