Abstract:
Majorizing sums of special form are constructed for rational functions and their derivatives R(μ)(z)R(μ)(z) (here μ=0,1,…μ=0,1,…, z∈C). As a consequence, several estimates of R(μ) in integral metrics are obtained on rectifiable curves Γ of finite density ω(Γ)=sup{mes1(Γ∩Δ)/diamΔ}, where the supremum is taken over all open discs Δ. Certain estimates on sets that are not necessarily connected are also obtained.
Citation:
V. I. Danchenko, “Several integral estimates of the derivatives of rational functions on sets of finite density”, Sb. Math., 187:10 (1996), 1443–1463
\Bibitem{Dan96}
\by V.~I.~Danchenko
\paper Several integral estimates of the~derivatives of rational functions on sets of finite density
\jour Sb. Math.
\yr 1996
\vol 187
\issue 10
\pages 1443--1463
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\crossref{https://doi.org/10.1070/SM1996v187n10ABEH000163}
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Linking options:
https://www.mathnet.ru/eng/sm163
https://doi.org/10.1070/SM1996v187n10ABEH000163
https://www.mathnet.ru/eng/sm/v187/i10/p33
This publication is cited in the following 12 articles:
F. G. Avkhadiev, I. R. Kayumov, S. R. Nasyrov, “Extremal problems in geometric function theory”, Russian Math. Surveys, 78:2 (2023), 211–271
A. D. Baranov, I. R. Kayumov, “Estimates for integrals of derivatives of $n$-valent functions and geometric properties of domains”, Sb. Math., 214:12 (2023), 1674–1693
A. D. Baranov, I. R. Kayumov, “Estimates for the
integrals of derivatives of rational functions in multiply connected
domains in the plane”, Izv. Math., 86:5 (2022), 839–851
A. D. Baranov, I. R. Kayumov, “Dolzhenko's inequality for $n$-valent functions: from smooth to fractal boundaries”, Russian Math. Surveys, 77:6 (2022), 1152–1154
A. D. Baranov, I. R. Kayumov, “Integral estimates of derivatives of rational functions in Hölder domains”, Dokl. Math., 106:3 (2022), 416–422
Akturk M.A., Lukashov A., “Sharp Markov-type inequalities for rational functions on several intervals”, J. Math. Anal. Appl., 436:2 (2016), 1017–1022
V. I. Danchenko, “Convergence of simple partial fractions in $L_p(\mathbb R)$”, Sb. Math., 201:7 (2010), 985–997
V. I. Danchenko, “Estimates of derivatives of simplest fractions and other questions”, Sb. Math., 197:4 (2006), 505–524
A. L. Lukashov, “Inequalities for derivatives of rational functions on several intervals”, Izv. Math., 68:3 (2004), 543–565
A. A. Pekarskii, “Bernstein type inequalities for arbitrary rational functions in the spaces $L_p$, $0<p<1$, on Lavrent'ev curves”, St. Petersburg Math. J., 16:3 (2005), 541–560
V. I. Danchenko, “Estimates of Green potentials. Applications”, Sb. Math., 194:1 (2003), 63–88
D. Ya. Danchenko, “On interpolation in the classes $E^p$”, Math. Notes, 66:3 (1999), 388–392
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