Subharmonic function

Definition

Let $U\subseteq\mathbb R^m$ be open. A function $u:U\to[-\infty,\infty)$ , not identically $-\infty$ on any connected component, is subharmonic if it is upper-semicontinuous and

u(x)\le \frac{1}{|S^{m-1}|}\int_{S^{m-1}}u(x+r\omega)\,d\omega

whenever the closed ball $\overline{B_r(x)}$ lies in $U$ . For $m=2$ , after identifying $\mathbb R^2\cong\mathbb C$ , this is the usual circular sub-mean inequality.

Distributional characterization

If $u\in L^1_{\mathrm{loc}}(U)$ , subharmonicity is equivalent to $\Delta u\ge0$ in the distributional sense. For $u\in C^2$ , this becomes the pointwise inequality $\Delta u\ge0$ .

Examples

Harmonic functions satisfy equality in the mean-value formula and are subharmonic. Every convex function on a convex open set is subharmonic. If $f$ is holomorphic, then $\log|f|$ is subharmonic, with its distributional Laplacian recording the zeros of $f$ .

References

Thomas Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995. DOI record.
David H. Armitage and Stephen J. Gardiner, Classical Potential Theory, Springer, 2001. DOI record.