On the failure of Bombieri's conjecture for univalent functions

Abstract

A conjecture of Bombieri [2] states that the coefficients of a normalized univalent function f should satisfy lim infRecently, Leung [10] disproved this conjecture for n = 2 and for all m ≥ 3 and, also, for n = 3 and for all odd m ≥ 5. Complementing his work we disprove it for all m > n ≥ 2 which are simultaneously odd or even and, also, for the case when m is odd, n is even and n ≤ m+1 2 . We mostly make use of trigonometry, but also employ Dieudonné's criterion for the univalence of polynomials.