---
tag: ZDHPQWWCQW
kind: concept
name: Harmonic majorant
msc: [31B05, 31A05]
sources:
  - url: https://en.wikipedia.org/wiki/Subharmonic_function
---

## Summary

A harmonic majorant of a function is a [[harmonic function|0DEB0JMN0W]] that lies above it everywhere on the domain.
The question whether a [[subharmonic function|Z1MHMGFNFW]] has a harmonic majorant measures how large it is near the boundary of its domain.
For example, a holomorphic function $f$ on the unit disc lies in the [[Hardy space|PNNNFJVF6W]] $H^p$ exactly when $\abs{f}^p$ has a harmonic majorant.
When a harmonic majorant exists, there is a smallest one, the [[least harmonic majorant|G0H97JZRS9]].

## Explanations

### L1 human

Picture the graph of a subharmonic function as a sagging sheet.
A harmonic majorant is a tight, perfectly balanced sheet stretched above it that never dips below it.
On the unit disc, the constant $1$ is a harmonic majorant of $\abs{z}^2$.
On the whole plane, $\abs{z}^2$ has no harmonic majorant at all, because it grows too fast.