I'm stuck at a Poisson integral problem and need some guidance.
Assume that the Poisson integral is known as $$\int\int_S P(r,r')u_0(r')dr'$$ and gives the solution to the boundary value problems $\Delta u(r)=0$ for |r|<1 and $u=u_0$ for |r|=1 and is harmonic in the ball $B=\{r\in \mathbb R^3:|r|<1\}$ and continuous up to the border $S=\{r\in \mathbb R^3 : |r|=1\}$
Show that $$\min_{r'\in S}u(r')\le u(r) \le\max_{r'\in S}u(r')$$ for all $r\in B$.
Appreciate any help.
Edit:
Got a hint from another source, but I still don't really understand how to solve it.
"For the first difference you can proceed from the obvious difference that $$\min_{r'\in S}u(r')\le u(r)$$ for all $r \in S$"