Riesz-Fej\'er inequalities for harmonic functions

TL;DR

This paper establishes Riesz-Fejér inequalities for complex harmonic functions in harmonic Hardy spaces, providing sharp results for certain p-values and variants for p=2, advancing harmonic analysis theory.

Contribution

It extends Riesz-Fejér inequalities to harmonic functions in Hardy spaces for all p > 1, including sharp bounds and variants for p=2.

Findings

Proved Riesz-Fejér inequality for harmonic functions in ${\bf h}^p$ for p > 1.

Established sharpness of the inequality for p in (1,2].

Derived two variants of the inequality for p=2.

Abstract

In this article, we prove the Riesz - Fej\'er inequality for complex-valued harmonic functions in the harmonic Hardy space ${\bf h}^p$ for all $p > 1$. The result is sharp for $p \in (1,2]$. Moreover, we prove two variant forms of Riesz-Fej\'er inequality for harmonic functions, for the special case $p=2$.