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 Ask… Show more
“…From the Grunsky inequalities (Henrici, 1986;Koepf, 2007) we know that, similar to (95), the inverse of the denominator in the right-hand side of (102) can be expanded into a Fourier series of w and z:…”
“…From the Grunsky inequalities (Henrici, 1986;Koepf, 2007) we know that, similar to (95), the inverse of the denominator in the right-hand side of (102) can be expanded into a Fourier series of w and z:…”
“…Finally, in 1985 de Branges [dB85] proved |a n | ≤ n for all n. For the historical development of the conjecture, see e.g. [Zor86] and [Koe07]. The coefficient problem for Σ appears to be even more difficult than for S. One reason is that there can be no single extremal function for all coefficients as the Koebe function.…”
Section: Quasiconformal Extensions Of Univalent Functionsmentioning
This survey article gives an account of quasiconformal extensions of univalent functions with its motivational background from Teichmüller theory and classical and modern approaches based on Loewner theory.
“…shows how that the coefficient bounds in Corollary 3.2 are related to the well-known Bieberbach conjecture [4] proved by de Branges in 1985 [6] (cf. [10]).…”
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