Subharmonic function

Summary

A subharmonic function is an upper semicontinuous function whose value at each point is at most its average over small circles around that point. Subharmonic functions play the role in two real dimensions that convex functions play in one. For a holomorphic function $f$, the functions $\log\abs{f}$ and $\abs{f}^p$ with $p > 0$ are subharmonic, which links complex analysis with potential theory. They are the main tool in Perron's method for the Dirichlet problem and in the study of growth of holomorphic functions.