caa a22 4500 463178262 CHVBK 20180406164831.0 cr unu---uuuuu 170326e20070601xx s 000 0 eng 10.1007/s11139-006-0244-2 doi (NATIONALLICENCE)springer-10.1007/s11139-006-0244-2 Koepf Wolfram Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, 34132, Kassel, Germany aut Bieberbach's conjecture, the de Branges and Weinstein functions and the Askey-Gasper inequality [Elektronische Daten] [Wolfram Koepf] The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [4]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [5] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [2] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [72] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated. Springer Science + Business Media, LLC, 2006 Bieberbach conjecture nationallicence Robertson conjecture nationallicence Milin conjecture nationallicence Convex functions nationallicence Starlike functions nationallicence Close-to-convex functions nationallicence Grunsky inequalities nationallicence Schiffer variation nationallicence Support points nationallicence Extreme points nationallicence Loewner differential equation nationallicence Loewner theory nationallicence Lebedev-Milin inequalities nationallicence de Branges theorem nationallicence de Branges functions nationallicence Weinstein functions nationallicence Hypergeometric functions nationallicence Generalized hypergeometric series nationallicence Askey-Gasper inequality nationallicence Askey-Gasper identity nationallicence Legendre addition theorem nationallicence FPS algorithm nationallicence Zeilberger algorithm nationallicence Maple nationallicence Symbolic computation nationallicence Computer algebra nationallicence The Ramanujan Journal Kluwer Academic Publishers 13/1-3(2007-06-01), 103-129 1382-4090 13:1-3<103 2007 13 11139 https://doi.org/10.1007/s11139-006-0244-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11139-006-0244-2 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Koepf Wolfram Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, 34132, Kassel, Germany aut NATIONALLICENCE 773 0- The Ramanujan Journal Kluwer Academic Publishers 13/1-3(2007-06-01), 103-129 1382-4090 13:1-3<103 2007 13 11139 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer