---
tag: 9ZV3T5C6B7
kind: definition
name: Subharmonic function
msc: [31A05, 31B05]
defines: Z1MHMGFNFW
uses: [02BH3MCBQB, 0DEB0JMN0W, 0E0B6RA8K0]
sources:
  - url: https://en.wikipedia.org/wiki/Subharmonic_function
---

## Definition

Let $\Omega$ be an [[open set|02BH3MCBQB]] in $\C$.
A function $u \colon \Omega \to [-\infty, \infty)$ is *subharmonic* if:

1. $u$ is upper semicontinuous, that is, $\set{z \in \Omega : u(z) < c}$ is open for every real $c$.
2. $u$ satisfies the sub-mean value inequality: for every $a \in \Omega$ and $r > 0$ with $\set{z : \abs{z - a} \le r} \subseteq \Omega$,
$$u(a) \le \frac{1}{2\pi}\int_0^{2\pi} u(a + r e^{it})\, dt.$$

## Remarks

Some authors also require that $u$ is not identically $-\infty$ on any connected component of $\Omega$.
The same definition, with averages over spheres, gives subharmonic functions on open sets of $\R^n$.
It is equivalent to require the inequality only for all small enough $r$ at each point.
Example: $u(z) = \abs{z}^2$ is continuous, and its mean over the circle of radius $r$ about $a$ is $\abs{a}^2 + r^2 \ge \abs{a}^2$, so $u$ is subharmonic.
Non-example: $u(z) = -\abs{z}^2$ is continuous, but its mean over the circle of radius $r$ about $0$ is $-r^2 < 0 = u(0)$, so clause 2 fails.
A function $u$ of class $C^2$ is subharmonic if and only if its [[Laplacian|0E0B6RA8K0]] satisfies $\Delta u \ge 0$.
A function is [[harmonic|0DEB0JMN0W]] if and only if both $u$ and $-u$ are subharmonic and $u$ is finite.
For a holomorphic function $f$ on $\Omega$, the function $\log\abs{f}$ is subharmonic, with value $-\infty$ at the zeros of $f$.