(a) Prove the following version of Green's theorem: suppose that $\Omega$ is a bounded region, bounded by finitely many $C^2$ simple closed curves. If $u$ and $v$ are twice continuously differentiable on $\bar{\Omega}$ then
$$
\int_{\partial \Omega}\left(u \frac{\partial v}{\partial \eta}-v \frac{\partial u}{\partial \eta}\right)|d z|=\int_{\Omega}(u \Delta v-v \Delta u) d x d y,
$$
where $\eta(\zeta)$ is the unit normal at $\zeta \in \partial \Omega$ pointing out of the region $\Omega,|d z|$ is arclength measure on $\partial \Omega$ and $d x d y$ is area measure on $\Omega$. If you already know a proof of some version of Green's theorem, you may use it to derive this version.
(b) Prove that (ii) implies (i) in Theorem 7.12 by directly verifying that the mean-value property holds. Hint: In $\mathbb{D} \backslash B(0, \varepsilon)$ set $v=\log |z|$, apply (a) and let $\varepsilon \rightarrow 0$.
(c) Similarly, show that if $v$ is twice continuously differentiable and $\Delta v \geq 0$ then $v$ is subharmonic because the mean-value inequality holds.
(d) If $g$ is differentiable on $\mathbb{R}$ and $g^{\prime \prime}>0$ then prove that $v\left(r e^{i t}\right)=g(\log r)$ is subharmonic.