Abstract. The classical Bernstein inequality estimates the derivative of a polynomial at a fixed point with the supremum norm and a factor depending on the point only. Recently, this classical inequality was generalized to arbitrary compact subsets on the real line. That generalization is sharp and naturally introduces potential theoretical quantities. It also gives a hint how a sharp $L^\alpha $ Bernstein inequality should look like. In this paper we prove this conjectured $L^\alpha $ Bernstein type inequality and we also prove its sharpness.

AMS Subject Classification (1991): 41A17, 26D05, 30C85

Keyword(s): polynomial inequalities, Bernstein inequality, potential theory, equilibrium measure

Received August 30, 2012, and in final form February 19, 2013. (Registered under 64/2012.)