In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.
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".
Formally, the definition can be stated as follows.
is called subharmonic if for any closed ball
Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.
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.
, in Euclidean space of dimension
is called a harmonic majorant of
This is called the Riesz representation theorem.
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
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.
is a subharmonic function if we define the value of
This observation plays a role in the theory of Hardy spaces, especially for the study of Hp when 0 < p < 1.
In the context of the complex plane, the connection to the convex functions can be realized as well by the fact that a subharmonic function
that is constant in the imaginary direction is convex in the real direction and vice versa.
Such an inequality can be viewed as a growth condition on
[1] Let φ be subharmonic, continuous and non-negative in an open subset Ω of the complex plane containing the closed unit disc D(0, 1).
The radial maximal function for the function φ (restricted to the unit disc) is defined on the unit circle by
If Pr denotes the Poisson kernel, it follows from the subharmonicity that
It can be shown that the last integral is less than the value at eiθ of the Hardy–Littlewood maximal function φ∗ of the restriction of φ to the unit circle T,
If f is a function holomorphic in Ω and 0 < p < ∞, then the preceding inequality applies to φ = |f |p/2.
It can be deduced from these facts that any function F in the classical Hardy space Hp satisfies
With more work, it can be shown that F has radial limits F(eiθ) almost everywhere on the unit circle, and (by the dominated convergence theorem) that Fr, defined by Fr(eiθ) = F(r eiθ) tends to F in Lp(T).
Subharmonic functions can be defined on an arbitrary Riemannian manifold.
Also, for twice differentiable functions, subharmonicity is equivalent to the inequality
[2] This article incorporates material from Subharmonic and superharmonic functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.