Bernstein type inequalities for rational functions on analytic curves and arcs

TL;DR

This paper extends Bernstein type inequalities for rational functions from the unit circle and real line to more general analytic Jordan arcs and curves, providing asymptotically sharp bounds.

Contribution

It introduces new sharp inequalities for rational functions on analytic Jordan arcs and curves, generalizing prior results on the unit circle and real line.

Findings

Established asymptotically sharp Bernstein inequalities on analytic Jordan arcs and curves.

Utilized Gonchar-Grigorjan estimates and Totik's polynomial constructions in proofs.

Extended classical inequalities to more general geometric settings.

Abstract

Borwein and Erd\'elyi proved a Bernstein type inequality for rational functions on the unit circle and on the real line. Here we establish asymptotically sharp extensions of their inequalities for rational functions on analytic Jordan arcs and curves. In the proofs key roles are played by Borwein-Erd\'elyi inequality on the unit circle, Gonchar-Grigorjan type estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials.