Bernstein- and Markov-type inequalities for rational functions

TL;DR

This paper establishes sharp Bernstein- and Markov-type inequalities for rational functions on smooth Jordan curves and arcs, generalizing polynomial inequalities and involving Green's functions with poles.

Contribution

It provides asymptotically sharp inequalities for rational functions on smooth curves, extending classical polynomial results to rational functions with poles.

Findings

Derived inequalities involve Green's functions with poles at rational function poles.

Results are asymptotically sharp for smooth Jordan curves and arcs.

Special case recovers polynomial inequalities when poles are at infinity.

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.