Subharmonic function

Remarks

Some authors also require that $u$ is not identically $-\infty$ on any connected component of $\Omega$. The same definition, with averages over spheres, gives subharmonic functions on open sets of $\R^n$. It is equivalent to require the inequality only for all small enough $r$ at each point. Example: $u(z) = \abs{z}^2$ is continuous, and its mean over the circle of radius $r$ about $a$ is $\abs{a}^2 + r^2 \ge \abs{a}^2$, so $u$ is subharmonic. Non-example: $u(z) = -\abs{z}^2$ is continuous, but its mean over the circle of radius $r$ about $0$ is $-r^2 < 0 = u(0)$, so clause 2 fails. A function $u$ of class $C^2$ is subharmonic if and only if its Laplacian satisfies $\Delta u \ge 0$. A function is harmonic if and only if both $u$ and $-u$ are subharmonic and $u$ is finite. For a holomorphic function $f$ on $\Omega$, the function $\log\abs{f}$ is subharmonic, with value $-\infty$ at the zeros of $f$.