Extremal Properties of Logarithmic Derivatives of Polynomials

Abstract

We study extremal properties of simple partial fractions ρn (i.e., the logarithmic derivatives of algebraic polynomials of degree n) on a segment and on a circle. We prove that for any a > 1 the poles of a fraction ρn whose sup norm does not exceed ln(1 + a−n) on [−1, 1] lie in the exterior of the ellipse with foci ±1 and sum of half-axes a. For a real-valued analytic function f bounded in the ellipse with $$ a=3+2\sqrt{2} $$ we show that if a real-valued simple partial fraction of order not greater than n is least deviating from f in the C([−1, 1])-metric, then such a fraction is unique and is characterized by an alternance of n + 1 points in the segment [−1, 1].