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Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality
Zusammenfassung
The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. 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 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] 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) [Weinstein1991] 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.
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@article{urn:nbn:de:hebis:34-200604038936,
author={Koepf, Wolfram},
title={Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality},
year={2005}
}
0500 Oax 0501 Text $btxt$2rdacontent 0502 Computermedien $bc$2rdacarrier 1100 2005$n2005 1500 1/eng 2050 ##0##urn:nbn:de:hebis:34-200604038936 3000 Koepf, Wolfram 4000 Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality / Koepf, Wolfram 4030 4060 Online-Ressource 4085 ##0##=u http://nbn-resolving.de/urn:nbn:de:hebis:34-200604038936=x R 4204 \$dPreprint 4170 Mathematische Schriften Kassel 7136 ##0##urn:nbn:de:hebis:34-200604038936
2006-04-03T09:00:40Z 2006-04-03T09:00:40Z 2005 urn:nbn:de:hebis:34-200604038936 http://hdl.handle.net/123456789/200604038936 208013 bytes application/pdf eng Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ Bieberbach-Vermutung de Branges and Weinstein Functions Askey-Gasper Inequality Bieberbachsche Vermutung 510 Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality Preprint The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. 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 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] 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) [Weinstein1991] 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. open access Koepf, Wolfram Mathematische Schriften Kassel 05, 13 Mathematische Schriften Kassel 05, 13
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