---
tag: G0H97JZRS9
kind: concept
name: Least harmonic majorant
msc: [31B05, 31A05]
specializes: [ZDHPQWWCQW]
sources:
  - url: https://en.wikipedia.org/wiki/Subharmonic_function
---

## Summary

The least harmonic majorant of a function is the smallest among its [[harmonic majorants|ZDHPQWWCQW]].
Let $u$ be a [[subharmonic function|Z1MHMGFNFW]] on a domain, with at least one harmonic majorant.
Then $u$ has a least harmonic majorant, which can be built as a limit of Poisson integrals over an exhausting sequence of subdomains.
Take a subharmonic function on a disc that is continuous up to the boundary.
Its least harmonic majorant is the Poisson integral of its boundary values.
It is used in the theory of Hardy spaces and, through its mirror image the greatest harmonic minorant, in the [[Riesz decomposition theorem|CBZ20SS8RQ]].

## Explanations

### L1 human

Among all balanced sheets lying above a sagging sheet, take the lowest one.
It touches the sagging sheet at the edge whenever possible and is pulled as tight as it can be.
For $\abs{z}^2$ on the unit disc, the lowest harmonic function above it is the constant $1$, which is the value of $\abs{z}^2$ on the boundary circle.