Application of a Bernstein type inequality to rational interpolation in the Dirichlet space

TL;DR

This paper establishes a Bernstein-type inequality for rational functions in the unit disc, relating Bergman and Hardy norms, with implications for rational interpolation in the Dirichlet space.

Contribution

It introduces a new Bernstein inequality involving Bergman and Hardy norms for rational functions with poles outside a scaled disk, advancing interpolation theory in the Dirichlet space.

Findings

Proved a Bernstein-type inequality for rational functions in the unit disc.

Connected Bergman and Hardy norms through this inequality.

Enhanced understanding of rational interpolation in the Dirichlet space.

Abstract

We prove a Bernstein-type inequality involving the Bergman and the Hardy norms, for rational functions in the unit disc \mathbb{D} having at most n poles all outside of \frac{1}{r}\mathbb{D}, 0