Bernstein inequality in L-alpha norms

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.