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Analogs of an inequality due to the Markov brothers for polynomials on a cube in \(\mathbb{R}^m\). (English. Russian original) Zbl 0903.26006
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 \(x\in Q\), \(Q=\{x\in\mathbb{R}^m:-1\leq x_i\leq 1,\;i=1,2,\ldots,n\}\).
26D05 Inequalities for trigonometric functions and polynomials
Full Text: DOI
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