Let $\Omega$ be a region in $\mathbb{C}$ and let $\phi:\Omega \to [-\infty,+\infty)$ be an upper semicontinuous function. TFAE: \ i)$\phi$ is subharmonic in $\Omega$,
ii) for any disc $\Delta_{z_0,r} \subset \subset \Omega$ we have $$ \phi(x_0) \leq (2\pi r)^{-1} \int_{\partial \Delta_{z_0,r}} \phi \; ds, $$ where ds is the element of arc,
iii) For any $\Delta_{z_0,r} \subset \subset \Omega$ we have $$ \phi(x_0) \leq (-2i\pi r^2)^{-1} \int_{ \Delta_{z_0,r}} \phi \; d\tau \wedge d\overline{\tau} $$
This is the seemingly awkward statement of a common characterization theorem one finds for subharmonic functions, from which I have omitted another equivalent condition because it wasn't relevant. I'm stuck in trying to prove that ii) implies iii). Any help will be appreciated.
Polar coordinates. For $0 < \rho \leqslant r$, we have
$$2\pi\rho \phi(z_0) \leqslant \int_{0}^{2\pi} \phi\left(z_0 + \rho e^{i\vartheta}\right)\,\rho\,d\vartheta,$$
by ii), and integrating that inequality from $0$ to $r$, we obtain
$$\begin{align} \pi r^2\phi(z_0) &= \int_0^r 2\pi \rho \phi(z_0)\,d\rho\\ &\leqslant \int_0^r \int_0^{2\pi} \phi\left(z_0 + \rho e^{i\vartheta}\right)\rho\,d\vartheta\,d\rho\\ &= \iint_{\Delta_{z_0\,r}} \phi(x+iy)\,dx\wedge dy\\ &= \frac{i}{2} \iint_{\Delta_{z_0\,r}} \phi(\tau) d\tau\wedge d\overline{\tau}. \end{align}$$