On the De Branges Theorem
Please always quote using this URN: urn:nbn:de:0297-zib-1765
- Recently, Todorov and Wilf independently realized that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. In this article, we present an elementary proof of this statement based on the defining differential equations system rather than the closed representation of de Branges' function system. Our proof does neither use special functions (like Wilf's) nor the residue theorem (like Todorov's) nor the closed representation (like both), but is purely algebraic. On the other hand, by a similar algebraic treatment, the closed representation of de Branges' function system is derived. Our whole contribution can be looked at as the study of properties of the Koebe function. Therefore, in a very elementary manner it is shown that the known proofs of the Bieberbach and Milin conjectures can be understood as a consequence of the Löwner differential equation, plus properties of the Koebe function.
Author: | Wolfram Koepf, Dieter Schmersau |
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Document Type: | ZIB-Report |
Date of first Publication: | 1995/03/24 |
Series (Serial Number): | ZIB-Report (SC-95-10) |
ZIB-Reportnumber: | SC-95-10 |
Published in: | Appeared in: Complex Variables 31 (1996) pp. 213-230 |