In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is below the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the boundary of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also inside the ball.

Superharmonic functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions.

Formal definition

Formally, the definition can be stated as follows. Let ${\displaystyle G}$ be a subset of the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ and let

be an upper semi-continuous function. Then, ${\displaystyle \varphi }$ is called subharmonic if for any closed ball ${\displaystyle {\overline {B(x,r)}}}$ of center ${\displaystyle x}$ and radius ${\displaystyle r}$ contained in ${\displaystyle G}$ and every real-valued continuous function ${\displaystyle h}$ on ${\displaystyle {\overline {B(x,r)}}}$ that is harmonic in ${\displaystyle B(x,r)}$ and satisfies ${\displaystyle \varphi (y)\leq h(y)}$ for all ${\displaystyle y}$ on the boundary ${\displaystyle \partial B(x,r)}$ of ${\displaystyle B(x,r)}$, we have ${\displaystyle \varphi (y)\leq h(y)}$ for all ${\displaystyle y\in B(x,r).}$

Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.

A function ${\displaystyle u}$ is called superharmonic if ${\displaystyle -u}$ is subharmonic.

Properties

• A function is harmonic if and only if it is both subharmonic and superharmonic.
• If ${\displaystyle \phi }$ is C² (twice continuously differentiable) on an open set ${\displaystyle G}$ in ${\displaystyle \mathbb {R} ^{n}}$, then ${\displaystyle \phi }$ is subharmonic if and only if one has ${\displaystyle \Delta \phi \geq 0}$ on ${\displaystyle G}$, where ${\displaystyle \Delta }$ is the Laplacian.
• The maximum of a subharmonic function cannot be achieved in the interior of its domain unless the function is constant, which is called the maximum principle. However, the minimum of a subharmonic function can be achieved in the interior of its domain.
• Subharmonic functions make a convex cone, that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic.
• The pointwise maximum of two subharmonic functions is subharmonic. If the pointwise maximum of a countable number of subharmonic functions is upper semi-continuous, then it is also subharmonic.
• The limit of a decreasing sequence of subharmonic functions is subharmonic (or identically equal to ${\displaystyle -\infty }$).
• Subharmonic functions are not necessarily continuous in the usual topology, however one can introduce the fine topology which makes them continuous.

Examples

If ${\displaystyle f}$ is analytic then ${\displaystyle \log |f|}$ is subharmonic. More examples can be constructed by using the properties listed above, by taking maxima, convex combinations and limits. In dimension 1, all subharmonic functions can be obtained in this way.

Riesz Representation Theorem

If ${\displaystyle u}$ is subharmonic in a region ${\displaystyle D}$, in Euclidean space of dimension ${\displaystyle n}$, ${\displaystyle v}$ is harmonic in ${\displaystyle D}$, and ${\displaystyle u\leq v}$, then ${\displaystyle v}$ is called a harmonic majorant of ${\displaystyle u}$. If a harmonic majorant exists, then there exists the least harmonic majorant, and

while in dimension 2, where ${\displaystyle v}$ is the least harmonic majorant, and ${\displaystyle \mu }$ is a Borel measure in ${\displaystyle D}$. This is called the Riesz representation theorem.

Subharmonic functions in the complex plane

Subharmonic functions are of a particular importance in complex analysis, where they are intimately connected to holomorphic functions.

One can show that a real-valued, continuous function ${\displaystyle \varphi }$ of a complex variable (that is, of two real variables) defined on a set ${\displaystyle G\subset \mathbb {C} }$ is subharmonic if and only if for any closed disc ${\displaystyle D(z,r)\subset G}$ of center ${\displaystyle z}$ and radius ${\displaystyle r}$ one has

Intuitively, this means that a subharmonic function is at any point no greater than the average of the values in a circle around that point, a fact which can be used to derive the maximum principle.

If ${\displaystyle f}$ is a holomorphic function, then