Inequalities of Rafalson type for algebraic polynomials

Abstract

For a positive Borel measure dmu, we prove that the constant gamma(n) (dv; dy) := (pi is an element ofPn\{0})sup integral-(infinity)(infinity) pi(2) (x)dmu(x)/<(∫-(infinity)(∞))over bar> pi(2) (x) dmu (x), can be represented by the zeros of orthogonal polynomials corresponding to dy in case (i) dv(x) = (A + Bx)dmu(x), where A + Bx is nonnegative on the support of dmu and (ii) dv(x) = (A + Bx(2))dmu(x), where dy is symmetric and A + Bx(2) is nonnegative on the support of dy. The extremal polynomials attaining the constant are obtained and some concrete examples are given including Markov-type inequality when dy is a measure for Jacobi polynomials. (C) 2004 Elsevier Inc. All rights reserved.