a subharmonic function

A harmonic function sits exactly at the average of its surroundings. A subharmonic function is one that sits at or below that average — it sags relative to the harmonic balance, like a stretched film weighed down from below or a temperature field with an interior heat sink. The prefix sub- means it lies under any harmonic function that matches it on a boundary, the way a chord lies under (or on) the curve of a convex graph.

There are two equivalent ways to pin this down. The averaging form: a continuous (more generally, upper semicontinuous) function u is subharmonic if for every small circle, the value at the center is less than or equal to the average of u over that circle — the reverse of the harmonic mean-value equality. The comparison form: u is subharmonic on D if, whenever h is harmonic on a subregion and u <= h on that subregion's boundary, then u <= h inside too. For smooth functions there is a third, very usable test: u is subharmonic exactly when its Laplacian is nonnegative, u_xx + u_yy >= 0. So subharmonic is to harmonic as convex is to linear: convex functions lie below their chords; subharmonic functions lie below their harmonic majorants.

Subharmonic functions are the raw material of the Perron method: because you can take the maximum of two subharmonic functions and still get a subharmonic function (unlike harmonic functions, which are too rigid to combine this way), they form a flexible family you can push upward toward a solution of the Dirichlet problem. They also satisfy their own maximum principle — a subharmonic function attains its maximum on the boundary — which is exactly what you would expect from something that bulges upward. A caution against a common slip: |f(z)| and log|f(z)| are subharmonic for holomorphic f, which is one source of the maximum-modulus principle, but they are generally not harmonic; subharmonic is a strictly larger, more flexible class than harmonic.