On the failure of Bombieri's conjecture for univalent functions

A conjecture of Bombieri states that the coefficients of a normalized univalent function $f$ should satisfy $$\underset{f\to K}{\mathrm{lim\hspace{0.17em}inf}}\frac{n-\mathrm{R}\mathrm{e}{a}_n}{m-\mathrm{R}\mathrm{e}{a}_m}=\underset{t\in \mathbb{R}}{\min }\frac{n\sin t-\sin (nt)}{m\sin t-\sin (mt)},$$ when $f$ approaches the Koebe function $K(z)=\frac{z}{(1-z{)}^2}$. Recently, Leung disproved this conjecture for $n=2$ and for all $m\ge 3$ and, also, for $n=3$ and for all odd $m\ge 5$. Complementing his work we disprove it for all $m>n\ge 2$ which are simultaneously odd or even and, also, for the case when $m$ is odd, $n$ is even and $n\le \frac{m+1}{2}$. We mostly make use of trigonometry, but also employ Dieudonn'e’s criterion for the univalence of polynomials.