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

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• 作者：Wolfram Koepf
• 关键词：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
• 刊名：The Ramanujan Journal
• 出版年：2007
• 出版时间：June, 2007
• 年：2007
• 卷：13
• 期：1-3
• 页码：103-129
• 全文大小：490 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Number Theory
  Field Theory and Polynomials
  Combinatorics
  Fourier Analysis
  Functions of a Complex Variable
  
• 出版者：Springer U.S.
• ISSN：1572-9303

文摘

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.