Let $P$ be a polynomial in $k$ variables with complex coefficients, and $\deg P=n$. If $k=1$ then there is Bernstein's inequality:$P'\le nP$, 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?

Tung, S. H. Bernstein's theorem for the polydisc. Proc. Amer. Math. Soc. 85 (1982), no. 1, 7376. 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 Bernsteintype inequality for the ball, Tung, S. H. Extension of Bernšteĭn's theorem. Proc. Amer. Math. Soc. 83 (1981), no. 1, 103106. MR0619992 (82k:32013) 

