Some results on the rational Bernstein Markov property in the complex plane

TL;DR

This paper investigates the rational Bernstein Markov property in the complex plane, comparing variants for rational functions with restricted poles and establishing conditions under which they imply polynomial BMP.

Contribution

It introduces and compares two variants of the rational BMP, providing sufficient conditions and a mass-density criterion for measures to satisfy the property.

Findings

Identifies conditions under which rational BMP variants imply polynomial BMP.

Provides a mass-density condition ensuring a measure satisfies rational BMP.

Establishes relationships between rational and polynomial BMP in the complex plane.

Abstract

The Bernstein Markov Property, shortly BMP, is an asymptotic quan- titative assumption on the growth of uniform norms of polynomials or rational functions on a compact set with respect to L {\mu} 2 -norms, where {\mu} is a positive finite measure. We consider two variants of BMP for rational functions with restricted poles and compare them with the polynomial BMP finding out some sufficient condi- tions for the latter to imply the former. Moreover, we recover a sufficient mass- density condition for a measure to satisfy the rational BMP on its support.