Journal of Convex Analysis 15 (2008), No. 2, 299--312
Copyright Heldermann Verlag 2008
An Application of the Krein-Milman Theorem to Bernstein and Markov Inequalities
Gustavo A. Muñoz-Fernández
Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain
gustavo_fernandez@mat.ucm.es
Yannis Sarantopoulos
Mathematics Department, National Technical University, Zografou Campus, 157 80 Athens, Greece
ysarant@math.ntua.gr
Juan B. Seoane-Sepúlveda
Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid
jseoane@mat.ucm.es
[Abstract-pdf]
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.
Keywords: Bernstein and Markov inequalities, trinomials, extreme points.
MSC: 41A17; 26D05
[ Fulltext-pdf (265 KB)] for subscribers only.