Some more counterexamples for Bombieri's conjecture on univalent functions

We disprove a conjecture of Bombieri regarding univalent functions in the unit disk in some previously unknown cases. The key step in the argument is showing that the global minimum of the real function $(n\sin x-\sin (nx))\mathrm{/}(m\sin x-\sin (mx))$ is attained at $x = 0$forintegers$m>n\geq2$when$m$isoddand$n$iseven,$m$ is sufficiently big and $0.5\le n\mathrm{/}m\le 0.8194$.