Geodesic
Markov's and Bernstein's Inequalities on Disjoint Intervals
Canadian journal of mathematics, Tome 33 (1981) no. 1, pp. 201-209

Voir la notice de l'article provenant de la source Cambridge University Press

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 establishedINEQUALITY 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). 
Borwein, Peter B. Markov's and Bernstein's Inequalities on Disjoint Intervals. Canadian journal of mathematics, Tome 33 (1981) no. 1, pp. 201-209. doi: 10.4153/CJM-1981-017-7
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[1] 1. Achieser, N. L., Theory of approximation (Ungar, New York, 1956). (Translated from the Russian.) Google Scholar

[2] 2. Bernstein, S. N., Collected works, Akad. Nauk SSSR, Moscow 11 (1954). Google Scholar

[3] 3. Lorentz, G. G., Approximation of functions (Holt, Rinehart and Winston, New York, 1966). Google Scholar

[4] 4. Markov, A. A., On a problem of D. I. Mendeleev, Izv. Akad. Nauk 62 (1889), 1–24. Google Scholar

[5] 5. Soble, A. B., Majorants of polynomial derivatives, Amer. Math. Monthly 64 (1957), 639–643. Google Scholar

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