Journal of Inequalities and Applications
Volume 4 (1999), Issue 4, Pages 315-325
doi:10.1155/S1025583499000429
An inequality for polynomials with elliptic majorant
Department of Mathematics, University of Sofia, boul. James Bourchier 5, Sofia 1164, Bulgaria
Received 21 November 1998; Revised 21 January 1999
Copyright © 1999 Geno Nikolov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let T¯(x):=Tn(ξx) be the transformed Chebyshev polynomial of the first kind, where ξ=cos(π/2n). We show here that T¯n has the greatest uniform norm in [−1,1] of its k-th derivative (k≥2) among all algebraic polynomials f of degree not exceeding n, which vanish at ±1 and satisfy the inequality |f(x)|≤1−ξ2x2 at the points {ξ−1cos((2j−1)π/2n−2)}j=1n−1.