Extremal Properties of Logarithmic Derivatives of Polynomials

We study extremal properties of simple partial fractions ρ n (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 ρ n 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 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].