Open Access

Wolfram Koepf, Dieter Schmersau

• \iffalse 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. \fi In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realized to be the same as de Branges'. In this article, we show how a variant of the Askey-Gasper identity can be deduced by a straightforward examination of Weinstein's functions which intimately are related with a Löwner chain of the Koebe function, and therefore with univalent functions.