Remez-, Nikolskii-, and Markov-Type Inequalities for Generalized Nonnegative Polynomials with Restricted Zeros

Generalized nonnegative polynomials were studied in a sequence of recent papers [4], [6], [7], [8], and [9]. A number of well-known inequalities in approximation theory were extended to them, by utilizing the generalized degree in place of the ordinary one. Our motivation was to find tools to examine systems of orthogonal polynomials simultaneously, associated with generalized Jacobi, or at least generalized nonnegative polynomial weight functions of degree at most F > 0. In a recent paper [10] we gave sharp estimates, in this spirit, for the Christoffel function on [ 1 , 1] and for the distances of the consecutive zeros of orthogonal polynomials associated with generalized nonnegative polynomial weight functions of degree at most F. Typically, the extension of a polynomial inequality to generalized nonnegative polynomials is not trivial, and the proof is far from a simple density argument. However, there is an inequalitY, the less known Remez inequality, which can be extended quite simply to generalized nonnegative polynomials, preserving at least the best possible order of magnitude. Based on this