Bernstein Polynomial Inequality on a Compact Subset of the Real Line

Abstract

We prove an analogue of the classical Bernstein polynomial inequality on a compact subset $E$ of the real line. The Lipschitz continuity of the Green function for the complement of $E$ with respect to the extended complex plane and the differentiability at a point of $E$ of a special, associated with $E$, conformal mapping of the upper half-plane onto the comb domain play crucial role in our investigation.