An Application of the Krein-Milman Theorem to Bernstein and Markov Inequalities

Abstract

Given a trinomial of the form

p(x)=ax^m+bx^n+c

with

a,b,c\in{\mathbb R}

, we obtain, explicitly, the best possible constant

\mathcal{M}_{m,n}(x)

in the inequality

|p'(x)| \le \mathcal{M}_{m,n}(x) \cdot \|p\|,

where

x\in[-1,1]

is fixed and

\|p\|

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

\mathcal{M}_{m,n}(x)

by means of classical convex analysis techniques, in particular, using the Krein-Milman approach.

Contact details are reproduced from the original publication and may be historical.

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

Mathematics Department, National Technical University, Zografou Campus, 157 80 Athens, Greece

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid

41A17; 26D05

R. M. Aron, M. Klimek: Supremum norms for quadratic polynomials, Arch. Math. 76 (2001) 73–80.https://doi.org/10.1007/s000130050544 Google Scholar
S. Bernstein: Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degré donné, Belg. Mem. in 4◦ 4 (1912) 1–103.Google Scholar
S. Bernstein: Collected works: Vol. I. The Constructive Theory of Functions (1905-1939), English translation, Atomic Energy Commission, Springfield (1958).Google Scholar
R. P. Boas: Inequalities for the derivatives of polynomials, Math. Mag. 42 (1969) 165–174.https://doi.org/10.1080/0025570X.1969.11975954 Google Scholar
A. A. Markov: On a problem of D. I. Mendeleev, Zap. Imp. Akad. Nauk. 62 (1889) 1–24 (in Russian).Google Scholar
A. A. Markov: On a question of D. I. Mendeleev, available at http://www.math.technion.ac.il/hat/papers.html.Google Scholar
G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda: Geometry of Banach spaces of trinomials, J. Math. Anal. Appl. 340 (2008) 1069–1087.https://doi.org/10.1016/j.jmaa.2007.09.010 Google Scholar
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).Google Scholar
E. V. Voronovskaja: The Method of Functional Calculus and its Applications, AMS, Providence (1970).https://doi.org/10.1090/mmono/028 Google Scholar

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.