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The Ramanujan Journal

, Volume 13, Issue 1–3, pp 103–129 | Cite as

Bieberbach’s conjecture, the de Branges and Weinstein functions and the Askey-Gasper inequality

  • Wolfram Koepf
Article

Abstract

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.

Keywords

Bieberbach conjecture Robertson conjecture Milin conjecture Convex functions Starlike functions Close-to-convex functions Grunsky inequalities Schiffer variation Support points Extreme points Loewner differential equation Loewner theory Lebedev-Milin inequalities de Branges theorem de Branges functions Weinstein functions Hypergeometric functions Generalized hypergeometric series Askey-Gasper inequality Askey-Gasper identity Legendre addition theorem FPS algorithm Zeilberger algorithm Maple Symbolic computation Computer algebra 

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© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KasselKasselGermany

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