Koornwinder, Tom H. A group theoretic interpretation of the last part of de Branges’ proof of the Bieberbach conjecture. (English) Zbl 0565.30011 Complex Variables, Theory Appl. 6, 309-321 (1986). A more conceptual and less computational proof is given for the last part of de Branges’ proof of the Bieberbach conjecture [cf. L. de Branges, Acta Math. 154, 137-152 (1985)], i.e. where the special functions enter and the Askey-Gasper inequality [cf. G. Gasper and R. Askey, Am. J. Math. 98, 709-737 (1976; Zbl 0355.33005)] is applied. General solutions of de Branges’ system of differential equations are brought in 1-1 correspondence first with Fourier-sine and next with spherical function expansions on the sphere \(S^ 3\). Restriction of spherical functions on \(S^ 5\) to \(S^ 3\) then finish the proof. Reviewer: Ch.F.Dunkl MSC: 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) Keywords:Milin conjecture; Gegenbauer polynomials; spherical functions on spheres; positive definite functions on spheres; Bieberbach conjecture; de Branges’ system of differential equations Biographic References: de Branges, Louis Citations:Zbl 0355.33005 PDFBibTeX XMLCite \textit{T. H. Koornwinder}, Complex Variables, Theory Appl. 6, 309--321 (1986; Zbl 0565.30011) Full Text: DOI Link