Geodesic
An Application of the Krein-Milman Theorem to Bernstein and Markov Inequalities
Journal of convex analysis, Tome 15 (2008) no. 2, pp. 299-312

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 41A17, 26D05
Mots-clés : Bernstein and Markov inequalities, trinomials, extreme points 
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     author = {G. A. Mu\~noz-Fern\'andez and Y. Sarantopoulos and J. B. Seoane-Sep\'ulveda},
     title = {An {Application} of the {Krein-Milman} {Theorem} to {Bernstein} and {Markov} {Inequalities}},
     journal = {Journal of convex analysis},
     pages = {299--312},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2008},
     url = {http://geodesic.mathdoc.fr/item/JCA_2008_15_2_JCA_2008_15_2_a7/}
}
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G. A. Muñoz-Fernández; Y. Sarantopoulos; J. B. Seoane-Sepúlveda. An Application of the Krein-Milman Theorem to Bernstein and Markov Inequalities. Journal of convex analysis, Tome 15 (2008) no. 2, pp. 299-312. http://geodesic.mathdoc.fr/item/JCA_2008_15_2_JCA_2008_15_2_a7/