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The de Branges theorem on univalent functions. (English) Zbl 0574.30018

In this article the authors present a simplified version of the de Branges proof of the Lebedev-Milin conjecture which implies the Robertson and the Bieberbach conjectures. The paper is well written and helps the reader to understand the remarkable work of de Branges. Some of the remarks of the end of the paper are of special interest. We mention in particular Remark 2, where the authors show that the weight functions \(\{\tau_ k\}\) made by de Branges can be motivated and shown to be unique in a certain sense. In Remark 3 the authors show that the application of the de Branges method is natural for the Lebedev-Milin conjecture, but cannot be applied naturally and directly even for the direct proof of \(| a_ 3| \leq 3\).
Reviewer: D.Aharonov

MSC:

30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C55 General theory of univalent and multivalent functions of one complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
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