# zbMATH — the first resource for mathematics

Bounds on the derivatives of polynomials on centrally symmetric convex bodies. (English. Russian original) Zbl 1102.41011
Izv. Math. 69, No. 3, 607-621 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 3, 179-192 (2005).
The author obtains multidimensional analogues for the Fréchet derivatives of polynomials on convex bodies in a pre-Hilbert space. The starting theorem that plays a key role in this paper gives the inequalities for derivatives of bidimensional polynomials of degree at most $$n$$ that map the square in the plane $$\mathbb{R}^2$$ to $$\mathbb{R}$$. The general multidimensional Markov inequalities are established as consequences of this particular case.

##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 46G25 (Spaces of) multilinear mappings, polynomials 52A40 Inequalities and extremum problems involving convexity in convex geometry
##### Keywords:
Markov type inequalities; multidimensional convex case
Full Text: