Sharp Riesz-Fej\'er inequality for harmonic Hardy spaces

TL;DR

This paper establishes a precise version of the Riesz-Fejér inequality for harmonic Hardy spaces on the unit disk, extending previous results and confirming a conjecture in the field.

Contribution

It provides the sharp form of the Riesz-Fejér inequality for harmonic Hardy spaces, resolving a previously posed conjecture and extending earlier work.

Findings

Proved the sharp Riesz-Fejér inequality for harmonic Hardy spaces

Extended the inequality to all p > 1 in the harmonic setting

Resolved the conjecture posed in prior research

Abstract

We prove sharp version of Riesz-Fej\'er inequality for functions in harmonic Hardy space $h^p(\mathbb{D})$ on the unit disk $\mathbb{D}$, for $p>1,$ thus extending the result from \cite{KPK} and resolving the posed conjecture.