Coefficients of powers of univalent functions

Abstract

Let S be the usual class of normalized univalent functions. We write and consider the behaviour of the upper bounds An (λ) of the |an(λ)| for fψ S. If f(z) is the Koebe function an (λ)= bn (λ) where For λ = 1 de Branges has recently proved Bieberbach's conjecture that An (λ)= bn (λ) and his method extends at once to the case λ > 1. The result is false for n = 2 and λ > 1. It is reasonable to compare An (λ) and bn (λ)for λ < 1. We show that the limit exists finitely for ¼ < λ < 1. We can give a characterization of K(λ) which enables lower bounds to be obtained numerically. From this it appears that if ¼ < λ < ,4998, then K(λ) > 1 We conjecture that KK(λ) = 1 for ½ ⩽ λ ⩽ 1, and K(λ) > 1, for λ < ½ The work is related to recent results of Baemstein and Pommerenke. If k = λ-1 is an integer, then the functions are the k-symmetric functions in S