Constants in Markov’s and Bernstein inequality on a finite interval in {mathbb {R}}

Abstract

In this paper we demonstrate the constants in the pointwise Bernstein inequality |P(α)(x)|≤2n(x-a)(b-x)α||P||[a,b],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} |P^{(\\alpha )}(x)|\\le \\left( \\frac{2n}{\\sqrt{(x-a)(b-x)}}\\right) ^{\\alpha }||P||_{[a,b]}, \\end{aligned}$$\\end{document}for the alpha -th derivative of an algebraic polynomial in L^{infty }-norms on an interval in {mathbb {R}}, where alpha ge 3. This result was obtained using the tools of theory of pluripotential and we apply it to get the main result which is a new generalization of V. A. Markov’s type inequalities ||P(α)||p≤C1/p2b-aα||Tn(α)||[-1,1]n2/p||P||p,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} ||P^{(\\alpha )}||_p\\le C^{1/{p}}\\left( \\frac{2}{b-a}\\right) ^{\\alpha }||T^{(\\alpha )}_{n}||_{[-1,1]}n^{2/p}||P||_{p}, \\end{aligned}$$\\end{document}for the alpha -th derivative of an algebraic polynomial in L^{p} norms, where pge 1. In particular, we show that for any alpha ge 3 the constant C in the V. A. Markov inequality satisfies the condition Cle 8left( frac{32cdot 3,94741cdot pi Malpha ^2}{3sqrt{3}}right) ^{1/p}.