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 Izv. Akad. Nauk SSSR Ser. Mat., 1979, Volume 43, Issue 2, Pages 277–293 (Mi izv1683)

An integral estimate for the derivative of a rational function

V. I. Danchenko

Abstract: Let there be given numbers $\alpha,q,\lambda,p$ and $n$, $0<\alpha<\infty$, $1\leqslant q\leqslant\infty$, $0<\lambda\leqslant\infty$, $1<p\leqslant\infty$, $n=1,2,…$, and let $R(n,p)$ be the class of rational functions $\rho(z)$ of degree $\leqslant n$, analytic for $|z|\leqslant1$, with
\begin{gather*} \|\rho\|_p=( \int_{|\zeta|=1}|\rho(\zeta)|^p |d\zeta|)^{1/p}\leqslant1
(\|\rho\|_\infty=\sup\{|\rho(z)|:|z|=1\}). \end{gather*}
It is proved that, if $\alpha\geqslant1+p^{-1}-q^{-1}$, then
$$\sup\{[ \int_0^1(1-r)^{\alpha\lambda-1}( \int_0^{2\pi}|\rho(r\cdot e^{i\varphi}|^q d\varphi)^{\lambda/q} dr]^{1/\lambda}:\rho\in R(n,p)\}<\infty.$$

Bibliography: 6 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1980, 14:2, 257–273

Bibliographic databases:

UDC: 517.5
MSC: 30E10, 41A20

Citation: V. I. Danchenko, “An integral estimate for the derivative of a rational function”, Izv. Akad. Nauk SSSR Ser. Mat., 43:2 (1979), 277–293; Math. USSR-Izv., 14:2 (1980), 257–273

Citation in format AMSBIB
\Bibitem{Dan79} \by V.~I.~Danchenko \paper An~integral estimate for the derivative of a~rational function \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1979 \vol 43 \issue 2 \pages 277--293 \mathnet{http://mi.mathnet.ru/izv1683} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=534594} \zmath{https://zbmath.org/?q=an:0443.30050|0413.30030} \transl \jour Math. USSR-Izv. \yr 1980 \vol 14 \issue 2 \pages 257--273 \crossref{https://doi.org/10.1070/IM1980v014n02ABEH001097} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1980KM96800003} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. 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
2. V. I. Danchenko, “Several integral estimates of the derivatives of rational functions on sets of finite density”, Sb. Math., 187:10 (1996), 1443–1463
3. J. Math. Sci. (N. Y.), 182:5 (2012), 639–645
4. Baranov A. Zarouf R., “A Bernstein-Type Inequality for Rational Functions in Weighted Bergman Spaces”, Bull. Sci. Math., 137:4 (2013), 541–556
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