Abstract. We prove Bernstein type inequalities for algebraic polynomials on the interval $I:=[-1,1]$ and for trigonometric polynomials on {\msbm R} when the roots of the polynomials are outside of a certain domain of the complex plane. The cases of real vs. complex coefficients are handled separately. In case of trigonometric polynomials with real coefficients and root restriction, the $L_p$-situation will also be considered. In most cases, the sharpness of the estimates will be shown.

AMS Subject Classification (1991): 41A17

Received February 26, 2001, and in revised form August 8, 2001. (Registered under 2835/2009.)