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

Abstract

Given $n\ge 1$ 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.