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Let $P$ be a polynomial in $k$ variables with complex coefficients, and $\deg P=n$. If $k=1$ then there is Bernstein's inequality:$||P'||\le n||P||$, where $||Q||=\max_{|z|=1} |Q(z)|$. So, are there similar results for $k\ge 2$? For example, what is the best $f(n,k)$, such that inequality $||\frac{\partial P}{\partial z_1}||+\cdots + ||\frac{\partial P}{\partial z_k}||\le f(n,k)||P||$, where $||Q||=\max_{|z_1|=\cdots=|z_k|=1}|Q(z_1,\cdots,z_k)|$, holds?

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I'm not sure. For example, if $P=z_1^{2n}+z_1^n z_2^n+z_2^{2n}$, then using univariate inequality I get that $||dP/dz_1||\le 2n ||P||$ and $||dP/dz_2||\le 2n ||P||$, so $||dP/dz_1||+||dP/dz_1||\le 4n ||P||$. But actually in that case $||dP/dz_1||=||dP/dz_2||=3n$ and $||P||=3$, so $||dP/dz_1||+||dP/dz_1||\le 2n||P||$ is valid. – Nurdin Takenov Jan 19 '10 at 15:02
up vote 2 down vote accepted

Tung, S. H. Bernstein's theorem for the polydisc. Proc. Amer. Math. Soc. 85 (1982), no. 1, 73--76. MR0647901 (83h:32017)

(from MR review): Let $P(z)$ be a polynomial of degree $N$ in $z=(z_1,\cdots,z_m)$; suppose that $|P(z)|\leq 1$ for $z\in U^m$; then $\|DP(z)\|\leq N$ for $z\in U^m$ where $\|DP(z)\|^2=\sum_{i=1}^m|\partial P/\partial z_i|^2$.

Here $U^m$ is the polydisc. Same author proved Bernstein-type inequality for the ball, Tung, S. H. Extension of Bernšteĭn's theorem. Proc. Amer. Math. Soc. 83 (1981), no. 1, 103--106. MR0619992 (82k:32013)

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