On the failure of Bombieri's conjecture for univalent functions

TL;DR

This paper demonstrates that Bombieri's conjecture for coefficients of normalized univalent functions fails in multiple cases, extending previous disproofs and employing trigonometry and univalence criteria for polynomials.

Contribution

It extends the disproof of Bombieri's conjecture to all cases where both indices are odd or even, and certain mixed cases, complementing prior work.

Findings

Disproves Bombieri's conjecture for all odd or even index pairs with n>2.

Shows the conjecture fails for specific mixed parity cases.

Utilizes trigonometry and Dieudonné's criterion in the proofs.

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.