L∞ Markov and Bernstein inequalities for Erdös weights

AbstractRecently, weighted Markov and Bernstein inequalities have been established for large classes of Freud weights, that is, weights of the form W(x) ≔e−Q(x), where Q(x) is even and of smooth polynomial growth at infinity. In this paper, we consider Erdős weights, which have the form W(x) ≔e−Q(x), where Q(x) is even and of faster than polynomial growth at infinity. For a large class of Erdős weights, we establish the Markov type inequality ‖P′‖R⩽CD′(an)‖PW‖R, (1) for n ⩾ 1 and P any polynomial of degree at most n. Here the norm is the sup norm, and C is independent of n and P, while an is the Mhaskar-Rahmanov-Saff number, that is, it is the positive root of the equation n = 2n∝01antQ′(ant)dt√1 − t2 For example, we consider Q(x) ≔expk(||α), where α > 0, and where expk denotes the kth iterated exponential, and give a more explicit formulation of (1). We also establish Bernstein type inequalities that for part of the range (− ∞, ∞) improve on (1)