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.
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
mathematical theorem ⓘ result in complex analysis ⓘ |
| alsoKnownAs | de Branges theorem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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}
NERFINISHED
ⓘ
functions obtained from the Koebe function by rotation ⓘ |
| extremalFunction |
Koebe function
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| namedAfter | Ludwig Bieberbach NERFINISHED ⓘ |
| partialResultsBy |
Aurel Wintner
NERFINISHED
ⓘ
Charles Loewner NERFINISHED ⓘ J. A. Jenkins NERFINISHED ⓘ Ludwig Bieberbach NERFINISHED ⓘ Menahem Schiffer NERFINISHED ⓘ Paul Koebe NERFINISHED ⓘ Y. Komatu NERFINISHED ⓘ Zeev Nehari NERFINISHED ⓘ |
| provedBy | Louis de Branges NERFINISHED ⓘ |
| relatedTo |
Koebe quarter theorem
NERFINISHED
ⓘ
Loewner differential equation NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.