Journal of Inequalities and Applications
Volume 4 (1999), Issue 4, Pages 315-325
doi:10.1155/S1025583499000429
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.