A Bernstein-type inequality for rational functions in weighted Bergman spaces

TL;DR

This paper establishes a sharp Bernstein-type inequality for rational functions with bounded poles in weighted Bergman spaces with polynomially decreasing weights, and shows the limitations for super-polynomial weights.

Contribution

It introduces an asymptotically sharp Bernstein inequality for rational functions in weighted Bergman spaces with polynomial weights, and demonstrates the boundary of its applicability.

Findings

Sharp Bernstein inequality for rational functions in polynomially weighted Bergman spaces

Limitations of the inequality for super-polynomially decreasing weights

Asymptotic behavior as the degree tends to infinity and r approaches 1

Abstract

Given $n\geq1$ and $r\in[0, 1),$ we consider the set $\mathcal{R}_{n, r}$ of rational functions having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D},$ were $\mathbb{D}$ is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in $\mathcal{R}_{n, r}\:$ (as n tends to infinity and r tends to 1-) in weighted Bergman spaces with "polynomially" decreasing weights. We also prove that this result can not be extended to weighted Bergman spaces with "super-polynomially" decreasing weights.