The de Branges theorem on univalent functions

Fitzgerald, Carl H.; Pommerenke, Ch.

doi:10.1090/s0002-9947-1985-0792819-9

Cited by 67 publications

(31 citation statements)

References 6 publications

(4 reference statements)

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“…Louis de Branges [2] - [3] proved the Bieberbach conjectures (2) and (3) for the functions (1). Fitzgerald and Pommerenke [4] - [5], Weinstein [21], and the author [15] - [19] proved in a simpler way the same conjecture, respectively (see the Grinshpan and Jahangiri reviews in [6] and [9], respectively).…”

mentioning

confidence: 84%

Another look at the De Branges weight functions

Todorov¹

2001

Analysis

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confidence: 84%

Another look at the De Branges weight functions

Todorov¹

2001

Analysis

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“…FitzGerald and Pommerenke [19] realized that the proof was independent of functional analysis and published a purely function theoretic version.…”

Section: The De Branges Theoremmentioning

confidence: 99%

“…To obtain (34), i.e. the hypergeometric representation (19) of τ n k , we multiply by K (z), and apply the FPS procedure a second time, this time w.r.t the variable z:…”

Section: The De Branges and The Weinstein Functionsmentioning

confidence: 99%

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

Koepf

2007

Ramanujan J

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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.

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“…Loewner Theory can be regarded as a theory providing a parametric representation of univalent functions in the unit disk D := {z : |z| < 1} based on an infinitesimal description of the semigroup of injective holomorphic self-maps of D. Originating in Loewner's paper [22] of 1923, this theory gave a great impact in the development of Complex Analysis, in which connection one might recall, e.g., its crucial role in the proof of the famous Bieberbach's Conjecture (see, e.g., [10,Chapter 17]) given by de Branges [11] in 1984, see also [12] for a simplified proof.…”

Section: Introductionmentioning

confidence: 99%

Loewner Theory in annulus II: Loewner chains

2011

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Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al. (J Reine Angew Math to appear, arXiv:0807.1594; Math Ann 344:947-962, 2009) and Contreras et al. (Revista Matemática Iberoamericana 26:975-1012, 2010) have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. This paper is devoted to the construction of a general version of Loewner Theory for the annulus launched in Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253). We introduce the general notion of a Loewner chain over a system of annuli and obtain a 1-to-1 correspondence between Loewner chains and evolution families in the doubly 352 M. D. Contreras et.alconnected setting similar to that in the Loewner Theory for the unit disk. Furthermore, we establish a conformal classification of Loewner chains via the corresponding evolution families and via semicomplete weak holomorphic vector fields. Finally, we extend the explicit characterization of the semicomplete weak holomorphic vector fields obtained in Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253) to the general case.

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The de Branges theorem on univalent functions

Abstract: We present a simplified version of the de Branges proof of the Lebedev-Milin conjecture which implies the Robertson and Bieberbach conjectures. As an application of an analysis of the technique, it is shown that the method could not be used directly to prove the Bieberbach conjecture.

Cited by 67 publications

References 6 publications

Another look at the De Branges weight functions

Another look at the De Branges weight functions

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

Loewner Theory in annulus II: Loewner chains

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