I would like to prove the strong maximum principle for sub harmonic functions:
if $u$ is $C^2$ in the ball of radius 1 surrounding origin and sub harmonic, attains a maximum at (0,0,0) then it must be constant in some ball centered at origin.
The method suggested is to consider $I(r) = 1/4\pi r^2 \int_ S udS$ where S is the boundary of the ball of radius r.
I have shown that the drivative of $I(r)$ w.r.t. r is $$1/4\pi r^2 \int_ S \nabla u d\vec S$$ and using this was able to show that $u(0) \leq I(r)$ for all $0 < r<1$. Any hints on how to use this to prove the strong maximum principle for subharmonic functions? Any help is greatly appreciated