Good evening I am doing exercises from a collection of problems and I have serious difficulties with this: Let $u:\mathbb{D}\rightarrow \mathbb{R}$ an harmonic function show that given $a \in \mathbb{D}$ $u$ takes the value $u(a)$ in $D(0,r)$ with $r>|a|$
From my point of view it should be very trivial, because if $r>|a|$ then $a \in D(0,r)$ and the function $u$ can valorate on this restriction. Is it enough to complete the exercise?
EDIT: $u$ takes the value $u(a)$ on $\partial D(0,r)$ with $r>|a|$
Thanks.
EDIT : (Solution)
If $u$ not takes the value $u(a)$ on $\partial D(0,r)$ then u(a) is an supremum or an infimum of the funcion on the frontier because $u$ is continous. By the principle of maximum module $|u(z)|\leq |u(a)| \forall z \in D(0,r)$ but $a\in D(0,r)$ then $u$ is constante on $\overline{D(0,r)}$ by the principle of identity $u$ should be constant on $\mathbb{D}$.