An application of the Krein-Milman theorem to Bernstein and Markov inequalities

Abstract

Given a trinomial of the form p(x) = ax(m) + bx(n) + c with a, b, c is an element of R, we obtain, explicitly, the best possible constant M.,,(x) in the inequality vertical bar p'(x)vertical bar <= M-m,M-n(x).parallel to p parallel to, where x is an element of [-1, 1] is fixed and parallel to p parallel to is the sup norm of p over [-1, 1]. This answers a question to an old problem, first studied by Markov, for a large family of trinomials. We obtain the mappings M-m,M-n(x) by means of classical convex analysis techniques, in particular, using the Krein-Milman approach.