Let $D(r)=\{(x,y):x^2+y^2\le r^2\}$, and let $u(x,y)$ be a non constant harmonic function in the unit disk $D(1)$. Let $\varphi(r)=\max_{(x,y)\in\partial D(r)}u(x,y)$ for $0\le r\le 1$.
Prove that
(a) $u$ is non constant on each disk $D(r)$ for all $0 <r<1$.
(b) $\varphi(r)$ is non decreasing in $[0,1]$.
(c) $\varphi (r)$ is strictly increasing in $[0,1]$.
My attempt:
(a) suppose $u$ is some constant $C$ in $D(r)$ for some $r<1$, let $S=\{x；u(x)=C\}$, then $S$ is closed in $D(1)$. Let $p\in S$, by $u$ is real analytic in $D(1)$, there exits $r$ such that u(x)=C in $B_r(p)$. Thus $S$ is open in $D(1)$. So $S=D(1)$, which is a contradiction.
For (b), by weak maximum principle for harmonic functions. Then it is obvious.
For (c), by strong maximum principle for harmonic functions. If $\varphi(r_1)=\varphi(r_2)$, where $r_2>r_1$. Then by strong maximum principle for harmonic functions, it is constant on $D(r_2)$, which is a contradiction with (a).
Could anyone help check if this is right? Thanks!