A proof of the Bieberbach conjecture

Abstract

In 1916, L. Bieberbach [2] conjectured that the inequality lanl--<.lallholds for every power series E~=~ a n z n with constant coefficient zero which represents a function with distinct values at distinct points of the unit disk.He also conjectured that equality holds with n>l only for a constant multiple of the Koebe functionwhere w is a constant of absolute value one.Bieberbach [2] verified the Bieberbach conjecture for the second coefficient.The Bieberbach conjecture for the third coefficient was verified by K. LOwner [9] in 1923.In 1955, P. R. Garabedian and M. Schiffer [7] verified the Bieberbach conjecture for the fourth coefficient.The Bieberbach conjecture for the sixth coefficient was verified in 1968 by R. N. Pederson [13] and, independently, by M. Ozawa [12].In 1972, Pederson and Schiffer [14] verified the Bieberbach conjecture for the fifth coefficient.No other case of the Bieberbach conjecture has previously been verified.A proof of the Bieberbach conjecture is now obtained for all remaining coefficients.Two other conjectures are also verified.In 1936, M. S. Robertson [17] conjectured that the inequality ~ml2+ ~212+... +~nl 2 ~< n~ll 2 holds for every odd power series E | ~ z 2n-~ which represents a function with distinct n= I/Jn values at distinct points of the unit disk.

Locations

• Acta Mathematica - View - PDF