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Algebra i Analiz, 2007, Volume 19, Issue 2, Pages 122–130 (Mi aa116)  

This article is cited in 9 scientific papers (total in 9 papers)

Research Papers

Estimates for derivatives of rational functions and the fourth Zolotarev problem

A. L. Lukashovab

a Saratov State University named after N. G. Chernyshevsky
b Department of Mathematics, Fatih University, Istanbul, Turkey

Abstract: An estimate is obtained for the derivatives of real rational functions that map a compact set on the real line to another set of the same kind. Many well-known inequalities (due to Bernstein, Bernstein—Szegö, V. S. Videnskii, V. N. Rusak, and M. Baran–V. Totik) are particular cases of this estimate. It is shown that the estimate is sharp. With the help of the solution of the fourth Zolotarev problem, a class of examples is constructed in which the estimates obtained turn into identities.

Keywords: Estimates of derivatives, optimal filter, Zolotarev problems.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:2, 253–259

Bibliographic databases:

MSC: Primary 53A04; Secondary 52A40, 52A10
Received: 11.10.2006

Citation: A. L. Lukashov, “Estimates for derivatives of rational functions and the fourth Zolotarev problem”, Algebra i Analiz, 19:2 (2007), 122–130; St. Petersburg Math. J., 19:2 (2008), 253–259

Citation in format AMSBIB
\by A.~L.~Lukashov
\paper Estimates for derivatives of rational functions and the fourth Zolotarev problem
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 2
\pages 122--130
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 2
\pages 253--259

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    This publication is cited in the following articles:
    1. V. N. Dubinin, S. I. Kalmykov, “A majoration principle for meromorphic functions”, Sb. Math., 198:12 (2007), 1737–1745  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. S. I. Kalmykov, “Majoration principles and some inequalities for polynomials and rational functions with prescribed poles”, J. Math. Sci. (N. Y.), 157:4 (2009), 623–631  mathnet  crossref  zmath
    3. V. N. Dubinin, D. B. Karp, V. A. Shlyk, “Izbrannye zadachi geometricheskoi teorii funktsii i teorii potentsiala”, Dalnevost. matem. zhurn., 8:1 (2008), 46–95  mathnet  elib
    4. V. N. Dubinin, “Emkosti kondensatorov i printsipy mazhoratsii v geometricheskoi teorii funktsii kompleksnogo peremennogo [Itogovyi nauchnyi otchet po mezhdistsiplinarnomu integratsionnomu proektu SO RAN: “Razrabotka teorii i vychislitelnoi tekhnologii resheniya obratnykh i ekstremalnykh zadach s prilozheniem v matematicheskoi fizike i gravimagnitorazvedke”]”, Sib. elektron. matem. izv., 5 (2008), 465–482  mathnet  mathscinet
    5. V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials”, Russian Math. Surveys, 67:4 (2012), 599–684  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. Totik V., “Bernstein-type inequalities”, J. Approx. Theory, 164:10 (2012), 1390–1401  crossref  mathscinet  zmath  isi
    7. A. V. Olesov, “Inequalities for majorizing analytic functions and their applications to rational trigonometric functions and polynomials”, Sb. Math., 205:10 (2014), 1413–1441  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Totik V., “Bernstein- and Markov-Type Inequalities For Trigonometric Polynomials on General Sets”, Int. Math. Res. Notices, 2015, no. 11, 2986–3020  crossref  mathscinet  zmath  isi  elib
    9. Lukashov A.L., Szabados J., “The order of Lebesgue constant of Lagrange interpolation on several intervals”, Period. Math. Hung., 72:2 (2016), 103–111  crossref  mathscinet  zmath  isi  elib  scopus
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