Chebyshev constants for the unit circle

TL;DR

This paper establishes a bound on the sum of inverse squared distances from a point on the unit circle to a system of points, characterizing when equality occurs, with two different proofs provided.

Contribution

It introduces a new inequality involving Chebyshev constants for points on the unit circle and provides two distinct proofs of this result.

Findings

The sum of inverse squared distances is bounded by n^2/4.

Equality holds iff the points are nth roots of unity.

Two different proof techniques are demonstrated.

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.