From Bieberbach's conjecture to de Branges' proof (Q1082474)

The author gives in this article a survey on the proof of the Bieberbach conjecture. Let f(z) be an analytic 1-1 function in the unit disc U s.t. \(f(z)=z+a_ 2z^ 2+a_ 3z^ 3+... \). Bieberbach conjectured in 1916 that \(| a_ n| \leq n\). This conjecture was finally settled in 1984 by de Branges. In this article the author presents the main partial results proved during the years, and also gives the main ideas and tools used by de Branges in his proof. It has to be pointed out that because of the great interest in de Branges theorem, this survey is one in a sequence of papers on the same subject by various authors. Recently the proof appeared in the book of \textit{P. Henrici} [Applied and computational complex analysis. III. (1986; Zbl 0578.30001)].