Markov's and Bernstein's Inequalities on Disjoint Intervals
Peter Borwein · Canadian Journal of Mathematics · 1981
In 1889, A. A. Markov proved the following inequality: INEQUALITY 1. (Markov [4]). If pn is any algebraic polynomial of degree at most n then where ‖ ‖A denotes the supremum norm on A. In 1912, S. N. Bernstein established INEQUALITY 2. (Bernstein [2]). If pn is any algebraic polynomial of degree at most n then for x ∈ (a, b). In this paper we extend these inequalities to sets of the form [a, b] ∪ [c, d]. Let Πn denote the set of algebraic polynomials with real coefficients of degree at most n. THEOREM 1. Let a < b ≦ c < d and let pn ∈ Πn. Then for x ∈ (a, b).