Subharmonic inequality 4

Question

Problem: If i have a subharmonic function $u$ on $D$ the unit disc, then for any two smaller radius $r_1\leq r_2$ we have $$\int_0^{2\pi}u(r_1e^{i\varphi})d\varphi\leq \int_0^{2\pi}u(r_2e^{i\varphi})d\varphi$$

Answer 1

$\newcommand\m[2]{\frac1{2\pi}\int_0^{2\pi}#1\left(#2 e^{it}\right)\,dt}$

Say $v$ is the solution to a certain Dirichlet problem: $v$ is harmonic in $|z|<r_2$ and $v(z)=u(z)$ for $|z|=r_2$. Then one of the basic properties of subharmonic functions says that $$u(z)\le v(z)\quad(|z|<r_2);$$in fact this is why they're called "subharmonic". So $$\begin{align}\m{u}{r_1}&\le\m{v}{r_1} \\&=v(0) \\&=\m{v}{r_2} \\&=\m{u}{r_2}.\end{align}$$