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=(z1,cdots,zm); suppose that |P(z)|leq1 for zinUm; then |DP(z)|leqN for zinUm where |DP(z)|2=summi=1|partialP/partialzi|2.
Here Um 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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