This is probably a very basic matter, but I am looking for a proof of the Poisson inequality for subharmonic functions, which reads

$$\varphi(r \mathrm{e}^{\mathrm{i} \theta})\leq\frac{1}{2\pi} \int_0^{2\pi}\mathrm{d} t\,P_r\left(\theta-t\right)\varphi\left(\mathrm{e}^{\mathrm{i} t}\right)$$

where the Poisson kernel is

$$P_r\left(\theta-t\right)\equiv\frac{1-r^2}{1-2r\cos\left(t-\theta\right)+r^2}.$$

Extensive search on the internet has not been very successful. Perhaps because this is trivial, but I don't see it. Of course, a fairly self-contained proof would be dynamite, but any help will be very much appreciated.

Thanks,

V.D.