On some Riesz and Carleman type inequalities for harmonic functions on the unit disk

TL;DR

This paper establishes new isoperimetric inequalities for harmonic functions in the unit disk within Hardy spaces, extending recent area-related results and discussing Riesz-type inequalities for holomorphic functions.

Contribution

It introduces novel isoperimetric inequalities for harmonic functions in Hardy spaces and extends existing results to complex harmonic functions and holomorphic functions.

Findings

Proved isoperimetric inequalities for harmonic functions in Hardy spaces.

Extended recent area-related inequalities to broader classes of harmonic functions.

Discussed Riesz-type inequalities for holomorphic functions.

Abstract

We prove some isoperimetric type inequalities for real harmonic functions in the unit disk belonging to the Hardy space $h^p$, $p>1$ and for complex harmonic functions in $h^4$. The results extend some recent results on the area. Further we discus some Riesz type results for holomorphic functions.