POLYNOMIALS LEAST DEVIATING FROM ZERO IN Lp (−1; 1), 0 ≤ p ≤ ∞, WITH A CONSTRAINT ON THE LOCATION OF THEIR ROOTS

Abstract

We study Chebyshev’s problem on polynomials that deviate least from zero with respect to Lp-means on the interval [−1; 1] with a constraint on the location of roots of polynomials. More precisely, we consider the problem on the set Pn(DR ) of polynomials of degree n that have unit leading coefficient and do not vanish in an open disk of radius R ≥ 1. An exact solution is obtained for the geometric mean (for p = 0) for all R ≥ 1; and for 0 < p < ∞ for all R ≥ 1 in the case of polynomials of even degree. For 0 < p < ∞ and R ≥ 1, we obtain two-sided estimates of the value of the least deviation. © 2023, Krasovskii Institute of Mathematics and Mechanics. All rights reserved.