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On the behaviour of constants in some polynomial inequalities

Volume 123 / 2019

Mirosław Baran, Leokadia Bialas-Ciez Annales Polonici Mathematici 123 (2019), 43-60 MSC: 32U20, 32U35, 41A17. DOI: 10.4064/ap180803-23-4 Published online: 20 September 2019


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 $E\subset \mathbb {C}^N$ we prove the inequalities $V(E)\le N C(E)$ and $H(E)\le \sqrt {N}\, V(E)$. Moreover, we calculate the Markov capacity for polydiscs and rectangular prisms in $\mathbb {C}^N$ 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.


  • Mirosław BaranFaculty of Mathematics, Physics and Technical Science
    Pedagogical University of Cracow
    Podchorążych 2
    30-084 Kraków, Poland
  • Leokadia Bialas-CiezInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland

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