Estimates on the Derivative of a Polynomial with a Curved Majorant Using Convex Techniques

Abstract

A mapping phi : [-1, 1] -> [0, infinity) is a curved majorant for a polynomial p in one real variable if vertical bar p(x)vertical bar <= phi(x) for all x is an element of [-1, 1]. If P(n)(phi)(R) is the set of all one real variable polynomials of degree at most n having the curved majorant phi, then we study the problem of determining, explicitly, the best possible constant M(n)(phi)(x) in the inequality vertical bar p'(x)vertical bar <= M(n)(phi)(x)parallel to p parallel to, for each fixed x is an element of [-1, 1], where p is an element of p(n)(phi)(R) and parallel to p parallel to is the sup norm of p over the interval [-1, 1]. These types of estimates are known as Bernstein type inequalities for polynomials with a curved majorant. The cases treated in this manuscript, namely phi(x) = root 1 - x(2) or phi(x) = vertical bar x vertical bar for an x is an element of [-1, 1] (circular and linear majorant respectively), were first studied by Rahman in [10]. In that reference the author provided, for each n is an element of N, the maximum of M(n)(phi)(x) over [-1, 1] as well as an upper bound for M(n)(phi)(x) for each x is an element of [-1, 1], where phi is either a circular or a linear majorant. Here we provide sharp Bernstein inequalities for some specific families of polynomials having a linear or circular majorant by means of classical convex analysis techniques (in particular we use the Krein-Milman approach).