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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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참고문헌 (4)
참고문헌 신청
D. Bshouty / 1987 / A variation of the Koebe mapping in a dense subset of S / Canad. J. Math 39(1):54~73
Iason Efraimidis / 2018 / On the Failure of Bombieri’s Conjecture for Univalent Functions / Computational Methods and Function Theory 18(3):427~438
Y. -J. Leung / 2016 / On the Bombieri numbers for the class S / J. Anal 24(2):229~250
D. Prokhorov / 2004 / On the local extremum property of the Koebe function / Math. Proc. Cambridge Philos. Soc 136(2):301~312
Iason Efraimidis / 2018 / On the Failure of Bombieri’s Conjecture for Univalent Functions / Computational Methods and Function Theory 18(3):427~438
Y. -J. Leung / 2016 / On the Bombieri numbers for the class S / J. Anal 24(2):229~250
D. Prokhorov / 2004 / On the local extremum property of the Koebe function / Math. Proc. Cambridge Philos. Soc 136(2):301~312
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