# Mean value properties of harmonic functions and related topics (a survey)

## TL;DR

This survey reviews mean value properties of harmonic, biharmonic, and metaharmonic functions, exploring how these properties characterize functions and how they can be weakened to identify harmonicity and other function classes.

## Contribution

It provides a comprehensive overview of mean value properties across different classes of functions and discusses conditions under which these properties imply harmonicity.

## Key findings

- Mean value properties characterize harmonic functions.
- Weakening mean value conditions can still imply harmonicity.
- Extensions to biharmonic and metaharmonic functions are discussed.

## Abstract

Results involving various mean value properties are reviewed for harmonic, biharmonic and metaharmonic functions. It is also considered how the standard mean value property can be weakened to imply harmonicity and belonging to other classes of functions.