Jackson’s inequality in the complex plane and the Łojasiewicz–Siciak inequality of Green’s function

We prove a generalization of Jackson’s inequality for 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 Łojasiewicz–Siciak inequality (ŁS). Moreover, we show that ŁS is a necessary condition for our Jackson type inequality. The resulting Jackson Property provides us with a large class of sets in the complex plane for which the global and local Markov inequalities are equivalent.