# Contractive inequalities for Hardy spaces

## TL;DR

This paper explores inequalities in Hardy spaces, focusing on coefficient bounds and the Riesz projection, proposing conjectures supported by numerical evidence.

## Contribution

It introduces new conjectures on contractive inequalities in Hardy spaces and provides numerical evidence supporting these conjectures.

## Key findings

- Numerical evidence supports the proposed conjectures.
- New inequalities relating to Hardy spaces are discussed.
- Insights into the Riesz projection as a map from L^q to H^p.

## Abstract

We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit disc. We study both coefficient estimates in terms of weighted $\ell^2$ sums and the Riesz projection viewed as a map from $L^q$ to $H^p$ with $q\ge p$. Some numerical evidence is given that supports our conjectures.