Extension of Bernstein's Theorem

where D,P(z) denotes the partial derivative of P with respect to z, at z for i = 1, ... ., m. A function H(z, w) defined for z, w E Cm with values in C is called a hermitian symmetric form if H(z, w) is linear in z for every fixed w and H(w, z) = H(z, w). In this paper, we extend the following complex version of Bernstein's theorem [5, p. 57] from a polynomial on the unit disk to a polynomial on the unit ball in several complex variables. Similar extensions of Bernstein's theorem and Markoff's theorem for polynomials of several real variables were obtained by Kellogg [4].