No abstract

Abstract. We prove a generalization of Dunham Jackson's famous approximation inequality to the case of compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e. the well known Hölder Continuity Property (HCP) and the less known but crucial Lojasiewicz-Siciak inequality ( LS). Moreover, we show that ( LS) is a necessary condition for our Jackson type inequality.

We prove that a compact subset of the complex plane satisfies a local Markov inequality if and only if it satisfies a Kolmogorov type inequality. This result generalizes a theorem established by Bos and Milman in the real case. We also show that every set satisfying the local Markov inequality is a sum of Cantor type sets which are regular in the sense of the potential theory.