Abstract:
Let $H_p$ be the Hardy space of functions $f$ that are analytic in the disk $|z|<1$ and let $J^\alpha f$ be the derivative of $f$ of order $\alpha$ in the sense of Weyl. It is shown, for example, that if $r$ is a rational function of degree $n\geqslant1$ with all its poles in the domain $|z|>1$, then $\|J^\alpha r\|_{H_\sigma}\leqslant cn^\alpha\|r\|_{H_p}$, where $p\in(1,\infty]$, $\alpha>0$, $\sigma=(\alpha+p^{-1})^{-1}$ and $c>0$ and depends only on $\alpha$ and $p$.
Bibliography: 32 titles.
Citation:
A. A. Pekarskii, “Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation”, Math. USSR-Sb., 52:2 (1985), 557–574
\Bibitem{Pek84}
\by A.~A.~Pekarskii
\paper Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation
\jour Math. USSR-Sb.
\yr 1985
\vol 52
\issue 2
\pages 557--574
\mathnet{http://mi.mathnet.ru/eng/sm2068}
\crossref{https://doi.org/10.1070/SM1985v052n02ABEH002906}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=754478}
\zmath{https://zbmath.org/?q=an:0609.41014|0567.41016}
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