Inverse Bernstein Inequalities and Min-max-min Problems on the Unit Circle

We give a short and elementary proof of an inverse Bernsteintype 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 with s > 0. 1. Inverse Bernstein-type inequality Inequalities involving the derivatives of polynomials often occur in approximation theory (see, e.g. [4], [6]). One of the most familiar of these inequalities is due to Bernstein which provides an upper bound for the derivative of a polynomial on the unit circle T of the complex plane. In [9], S. Khrushchev derived a rather striking inverse Bernstein-type inequality, a slight improvement of which may be stated as follows.