A bivariate Markov inequality for Chebyshev polynomials of the second kind

This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov's theorem to real normed linear spaces as an easy corollary. To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel-Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials. (C) 2011 Elsevier Inc. All rights reserved.