Let $U\subseteq \mathbb{C}$ be an open set and let $u(z)=-\log(\mathrm{dist}(z,\partial U))$.
I need to show that $u$ is subharmonic on $U$?
$\partial U$ it means the boundary of $U$.
Let $U\subseteq \mathbb{C}$ be an open set and let $u(z)=-\log(\mathrm{dist}(z,\partial U))$.
I need to show that $u$ is subharmonic on $U$?
$\partial U$ it means the boundary of $U$.
Hint: For fixed $x\in\partial U$, the function $-\log(|z-x|)$ is harmonic on $U$.
Suppose $U$ is connected open. Fix a $z\in U.$ Since $U$ is open, there is an $s>0$ such that $B(z,s)\subset U.$ Now, $\frac{1}{|\partial B(z,s)|}\int_{\partial B(z,s)}dist(w,\partial U) \leq dist(w,\partial U).$ Now, use Jensen inequality to finish the proof. Above argument shows that your function, say f(w) which is defined in the interior of U is subharmonic. You should also discuss, for instance, the riemann integrability of $dist(w,\partial U)$ on the boundary set $\partial B(z,r)$ if you want a complete solution.