Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

Annales Polonici Mathematici

Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

On the behaviour of constants in some polynomial inequalities

Tom 123 / 2019

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

Streszczenie

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.

Autorzy

• Mirosław BaranFaculty of Mathematics, Physics and Technical Science
Pedagogical University of Cracow
Podchorążych 2
30-084 Kraków, Poland
e-mail