Acta Mathematica

Volume 219 (2017)

Number 1

Bernstein- and Markov-type inequalities for rational functions

Pages: 21 – 63

DOI: https://dx.doi.org/10.4310/ACTA.2017.v219.n1.a3

Authors

Sergei Kalmykov (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China; and Far Eastern Federal University, Vladivostok, Russia)

Béla Nagy (MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Hungary)

Vilmos Totik (MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Hungary; and Department of Mathematics and Statistics, University of South Florida, Tampa, Fl., U.S.A.)

Abstract

Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on $C^2$ smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green’s functions with poles at the poles of the rational functions in question. As a special case (when all the poles are at infinity) the corresponding results for polynomials are recaptured.

Full Text (PDF format)

Received 9 July 2016

Accepted 20 June 2017

Published 31 January 2018