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arXiv · 1006.5153

Chebyshev constants for the unit circle

Gergely Ambrus·Keith M. Ball·T. Erdélyi
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Abstract

It is proven that for any system of n points z_1, ..., z_n on the (complex) unit circle, there exists another point z of norm 1, such that $$\sum 1/|z-z_k|^2 \leq n^2/4.$$ Equality holds iff the point system is a rotated copy of the nth unit roots. Two proofs are presented: one uses a characterisation of equioscillating rational functions, while the other is based on Bernstein's inequality.

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Gergely Ambrus, Keith M. Ball, T. Erdélyi. 2011-11-08. Chebyshev constants for the unit circle. https://doi.org/10.1112/blms%2Fbds082

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