Publication: An application of the Krein-Milman theorem to Bernstein and Markov inequalities
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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.
R. M. Aron, M. Klimek: Supremum norms for quadratic polynomials, Arch. Math. 76 (2001) 73–80. S. Bernstein: Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degree donné, Belg. Mem. in 4◦ 4 (1912) 1–103. S. Bernstein: Collected works: Vol. I. The Constructive Theory of Functions (1905-1939), English translation, Atomic Energy Commission, Springfield (1958). R. P. Boas: Inequalities for the derivatives of polynomials, Math. Mag. 42 (1969) 165–174. A. A. Markov: On a problem of D. I. Mendeleev, Zap. Imp. Akad. Nauk. 62 (1889) 1–24(in Russian). A. A. Markov: On a question of D. I. Mendeleev, available at http://www.math.technion. ac.il/hat/papers.html. G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda: Geometry of Banach spaces of trinomials, J. Math. Anal. Appl. 340 (2008) 1069–1087. E. V. Voronovskaja: The functional of the first derivative and improvement of a theorem of A. A. Markov, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959) 951–962 (in Russian). E. V. Voronovskaja: The Method of Functional Calculus and its Applications, AMS, Providence (1970).