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arXiv · 1710.10462

Some more counterexamples for Bombieri's conjecture on univalent functions

Iason Efraimidis·Carlos Pastor
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Abstract

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 $\big(n\sin{x}-\sin(nx)\big)/\big(m\sin{x}-\sin(mx)\big)$ is attained at $x = 0$ for integers $m>n\geq2$ when $m$ is odd and $n$ is even, $m$ is sufficiently big and $0.5 \leq n/m \leq 0.8194$.

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Iason Efraimidis, Carlos Pastor. 2017-10-28. Some more counterexamples for Bombieri's conjecture on univalent functions. https://arxiv.org/abs/1710.10462

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