Analogs of an inequality due to the Markov brothers for polynomials on a cube in \(\mathbb{R}^m\). (English. Russian original) Zbl 0903.26006

Skalyga, V. I.

Math. Notes 60, No. 5, 589-593 (1996); translation from Mat. Zametki 60, No. 5, 783-787 (1996).

The main result is the following extension of A. A. and V. A. Markov’s inequality to the multivariate case: \[ \max_{1\leq i,j\leq m}\left|\frac{\partial^2P_n(x)}{\partial x_i\partial x_j}\right|\leq\frac{n^2(n^2-1)}{3},\;x\in\mathbb{Q}, \] \[ \left(\sum_{i=1}^m\left(\frac{\partial P_n(x)}{\partial x_i}\right)^2\right)^{1/2}\leq n^2,\;x\in\mathbb{Q}, \] for any polynomial \(P_n\) of degree \(n\) in \(m\) variables such that \(| P_n(x)|\leq 1\) for \(x\in Q\), \(Q=\{x\in\mathbb{R}^m:-1\leq x_i\leq 1,\;i=1,2,\ldots,n\}\).

Reviewer: A.Lukashov (Saratov)

MSC:

26D05

Inequalities for trigonometric functions and polynomials

Keywords:

Markov inequality; several variables

References:

[1]	S. N. Bernshtein,Collected Works [in Russian], Vol. 2, Akad. Nauk SSSR, Moscow (1954).
[2]	O. D. Kellog,Math. Zeit., 55–66 (1927).
[3]	R. Don,J. Approx. Theory.,11, No. 3 (1974).
[4]	L. V. Kantorovich and G. P. Akilov,Functional Analysis in Normed Spaces [in Russian], Fizmatgiz, Moscow (1959). · Zbl 0127.06102
[5]	M. I. Ganzburg,Uspekhi Mat. Nauk [Russian Math. Surveys],34, No. 1, 225–226 (1979).