Let $u(x,y), v(x,y)$ be harmonic functions in $\mathbb{R}^2$. I want to find the sign of the difference $u(0,0)-v(0,0)$ given that $u|_{x^2+y^2=1}=\frac{1}{2}-\sin{\phi}$, where $x= \cos{\phi}, y=\sin{\phi}, \phi \in [0,2 \pi)$ and $v|_{(x-2)^2+y^2=3^2} \leq 0$.

Applying the theorem for the solution of the Dirichlet problem defined on a sphere I found that $u(0,0)=\frac{1}{2}$.

In order to find something about $v(0,0)$ I thought to use the following:

If $u \in C^0(\overline{\Omega})$ is harmonic in $\Omega$ and $u \leq 0$ in $\partial{\Omega}$ then $u \leq 0$ in $\overline{\Omega}$.