On the failure of Bombieri's conjecture for univalent functions

Abstract

A conjecture of Bombieri states that the coefficients of a normalized univalent function $f$ should satisfy $$ \liminf_{f\to K} \frac{n-{\rm Re\,}a_n}{m-{\rm Re\,}a_m} = \min_{t\in{\mathbb R}} \, \frac{n\sin t -\sin(nt)}{m\sin t -\sin(mt)}, $$ when $f$ approaches the Koebe function $K(z)=\frac{z}{(1-z)^2}$. Recently, Leung disproved this conjecture for $n=2$ and for all $m\geq3$ and, also, for $n=3$ and for all odd $m\geq5$. Complementing his work we disprove it for all $m>n\geq2$ which are simultaneously odd or even and, also, for the case when $m$ is odd, $n$ is even and $n\leq \frac{m+1}{2}$. We mostly make use of trigonometry, but also employ Dieudonn\'e's criterion for the univalence of polynomials.