Reflections on a theorem of V. Andrievskii

For a given point of a compact subset E of the real line four properties are proven to be pairwise equivalent: local Bernstein-inequality, local higher-order Bernstein-inequality, local Lip 1 continuity of the Green's function and local Lip 1 property of the equilibrium measure. Furthermore, in connection with a result of V. Andrievkskii, it is shown that these equivalent properties are closely related to Bernstein's approximation theorem and its generalization given by R. K. Vasiliev. Similar results are established at endpoints of subintervals of E, where the local Bernstein-inequality is replaced by the local Markov-inequality and Lip 1 is replaced by Lip 1/2.