Given a nonnegative smooth function $g$ with compact support in the complex plane (e.g. a mollifier) and a subharmonic function $u$ (defined on a sufficiently large domain in $\mathbb C$), I would like to legitimize the use of Fatou's lemma in the following case: $$ \limsup_{s\to0}\int_{\mathbb C}\frac{4(C_u(z,s)-u(z))}{s^2}g(z)d\sigma(z)\leq\int_{\mathbb C}\limsup_{s\to0}\frac{4(C_u(z,s)-u(z))}{s^2}g(z)d\sigma(z), $$ where $$ C_u(z,s):=\frac{1}{2\pi}\int_0^{2\pi}u(z+se^{it})dt $$ is the mean value of $u$ of the circle with center $z$ and radius $s$. The integration ($\sigma(z)$) is meant in the (two dimensional) Lebesgue sense.
I know that subharmonic functions (in my case only upper semicontinuous) are bounded above on compact sets, and that the (reverse) Fatou lemma is true if I could show that $$ \frac{4(C_u(z,s)-u(z))}{s^2} $$ is bounded above by an integrable function, but I have no idea how to prove that. I can assume, in addition, that $$ z\mapsto\limsup_{s\to0}\frac{4(C_u(z,s)-u(z))}{s^2} $$ is subharmonic as well. Does anyone know if the above statement is true, and if so, how to prove it? Or does there exist a counterexample? I would be very grateful if someone could help me out. Thanks in advance!