# subharmonic and superharmonic functions

First let’s look at the most general definition.

###### Definition.

Let $G\subset {\mathbb{R}}^{n}$ and let $\phi :G\to \mathbb{R}\cup \{-\mathrm{\infty}\}$ be an upper semi-continuous function,
then $\phi $ is subharmonic if for every $x\in G$ and $r>0$ such that
$\overline{B(x,r)}\subset G$ (the closure of the open ball of radius $r$ around $x$ is still in $G$) and every real valued continuous function^{} $h$ on
$\overline{B(x,r)}$ that is harmonic in $B(x,r)$ and satisfies $\phi (x)\le h(x)$
for all $x\in \partial B(x,r)$ (boundary of $B(x,r)$) we have that
$\phi (x)\le h(x)$ holds for all $x\in B(x,r)$.

Note that by the above, the function which is identically $-\mathrm{\infty}$ is subharmonic, but some authors exclude this function by definition. We can define superharmonic functions in a similar fashion to get that $\phi $ is superharmonic if and only if $-\phi $ is subharmonic.

If we restrict our domain to the complex plane^{} we can get the following definition.

###### Definition.

Let $G\subset \u2102$ be a region and let $\phi :G\to \mathbb{R}$ be a continuous function. $\phi $ is said to be subharmonic if whenever $D(z,r)\subset G$ (where $D(z,r)$ is a closed disc around $z$ of radius $r$) we have

$$\phi (z)\le \frac{1}{2\pi}{\int}_{0}^{2\pi}\phi (z+r{e}^{i\theta})\mathit{d}\theta ,$$ |

and $\phi $ is said to be superharmonic if whenever $D(z,r)\subset G$ we have

$$\phi (z)\ge \frac{1}{2\pi}{\int}_{0}^{2\pi}\phi (z+r{e}^{i\theta})\mathit{d}\theta .$$ |

Intuitively what this means is that a subharmonic function is at any point
no greater than the average^{} of the values in a circle around that point. This implies that a non-constant subharmonic function does not achieve its maximum
in a region $G$ (it would achieve it at the boundary if it is continuous there). Similarly for a superharmonic
function, but then a non-constant superharmonic function does not achieve its
minumum in $G$. It is also easy to see that $\phi $ is subharmonic if and only if $-\phi $ is superharmonic.

Note that when equality always holds in the above equation then $\phi $ would in fact be a harmonic function. That is, when $\phi $ is both subharmonic and superharmonic, then $\phi $ is harmonic.

It is possible to relax the continuity statement above to take $\phi $ only upper semi-continuous in the subharmonic case and lower semi-continuous in the superharmonic case. The integral will then however need to be the Lebesgue integral (http://planetmath.org/Integral2) rather than the Riemann integral which may not be defined for such a function. Another thing to note here is that we may take ${\mathbb{R}}^{2}$ instead of $\u2102$ since we never did use complex multiplication. In that case however we must rewrite the expression $z+r{e}^{i\theta}$ in of the real and imaginary parts to get an expression in ${\mathbb{R}}^{2}$.

It is also possible generalize the range of the functions as well. A subharmonic function could have a range of $\mathbb{R}\cup \{-\mathrm{\infty}\}$
and a superharmonic function could have a range of $\mathbb{R}\cup \{\mathrm{\infty}\}$. With this generalization^{}, if $f$ is a holomorphic function^{}
then $\phi (z):=\mathrm{log}|f(z)|$ is a subharmonic function if we
define the value of $\phi (z)$ at the zeros of $f$ as $-\mathrm{\infty}$.
Again it is important to note that with this
generalization we again must use the Lebesgue integral.

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | subharmonic and superharmonic functions |

Canonical name | SubharmonicAndSuperharmonicFunctions |

Date of creation | 2013-03-22 14:19:39 |

Last modified on | 2013-03-22 14:19:39 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 12 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 31C05 |

Classification | msc 31A05 |

Classification | msc 31B05 |

Related topic | HarmonicFunction |

Defines | subharmonic |

Defines | subharmonic function |

Defines | superharmonic |

Defines | superharmonic function |