On the asymptotic Bieberbach conjecture

Abstract

The set S consists of complex functions f, univalent in the open unit disk, with f(0)=f'(0)-1=0. We use the asymptotic behavior of the positive semidefinite FitzGerald matrix to show that there is an absolute constant N0 such that, for any f(z)=z+?n=28anzn?S with |a3|=2.58, we have |an|<n for all n>N0.