Let $K \subset \mathbb C$ be compact, and let $u(z) = -\log(\mathrm{dist}(z,K))$ be defined on $\mathbb C \setminus K$. May I get a hint for proving that $u(z)$ is subharmonic?
Subharmonic, here, is defined as u.s.c. and also satisfies the sub-mean-value property. For $C^2$ functions, $f$ is subharmonic $\iff \Delta f \geq 0$.
Current work:
Set $U = \mathbb C \setminus K$. For a $w \in \mathbb \partial U$, we set $u_w(z) = -\log|z-w|$. Then $u_w$ can be shown to be harmonic (it's rather straightforward to compute the Laplacian). Moreover, it is clear that $\displaystyle u(z) = \inf_{w \in \partial U}u_w(z)$. But I'm having trouble showing that $u$ is therefore subharmonic. I have the feeling I'm just missing something trivial.