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?



Bernstein inequality for multivariate polynomial
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)


