Asymptotic sharpness of a Bernstein-type inequality for rational functions in H^{2}

TL;DR

This paper establishes the asymptotic sharpness of a Bernstein-type inequality for rational functions in the Hardy space H^{2}, focusing on functions with poles outside a scaled disk, advancing understanding of their boundary behavior.

Contribution

It proves the asymptotic sharpness of a Bernstein-type inequality for rational functions in H^{2} with poles outside a scaled disk, extending previous results.

Findings

The inequality is asymptotically sharp for large n.

Provides bounds for rational functions with poles outside a scaled disk.

Enhances understanding of boundary behavior of rational functions in H^{2}.

Abstract

A Bernstein-type inequality in the standard Hardy space H^{2} of the unit disc \mathbb{D}=\{z\in\mathbb{C}:\,|z|<1\}, for rational functions in \mathbb{D} having at most n poles all outside of \frac{1}{r}\mathbb{D}, 0