  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.

Complex Variables