On the behaviour of constants in some polynomial inequalities

Abstract

We study the asymptotical behaviour of optimal constants in the Hölder continuity property (HCP) of the Siciak extremal function and in the Vladimir Markov inequality equivalent to HCP. We observe that the optimal constants in polynomial inequalities of Markov and Bernstein type are related to some quantities that resemble capacities. We call them Hölder’s and Markov’s capacity and denote by H(E), V (E) respectively. We compare these two capacities with the L-capacity C(E). In particular, for any compact set E ⊂ CN we prove the inequalities V (E) ≤ NC(E) and H(E) ≤√N V (E). Moreover, we calculate the Markov capacity for polydiscs and rectangular prisms in CN and we find that in these cases V (E) = H(E) = C(E). Additionally, some new conditions equivalent to HCP and to the Andrey Markov inequality are given.