Inverse Bernstein inequalities and min-max-min problems on the unit circle

TL;DR

This paper presents an elementary proof of an inverse Bernstein inequality for polynomials with zeros on the unit circle and demonstrates the optimality of equally-spaced points for certain potential minimax problems, extending to Riesz potentials.

Contribution

Provides a new elementary proof of an inverse Bernstein inequality and extends the optimality of equally-spaced points to a broad class of potentials including Riesz potentials.

Findings

Equally-spaced points solve the min-max-min problem for logarithmic potential.

The inverse Bernstein inequality applies to polynomials with zeros on the unit circle.

Optimality extends to Riesz potentials with s>0.

Abstract

We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min-max-min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials $1/r^s$ with $s>0.$