Is it possible to find a superharmonic function with such conditions

Question

$\lim_{z\rightarrow 0}\psi(z) = 0$, $\lim_{z\rightarrow w}\psi(z) = 1$ for $\{w: |w|=r, w\neq r\}$ and $\psi$ is superharmonic on $B(0;r)\setminus [0,\infty)$? In the context I use, a superharmonic function has to have super mean value property. (i.e. it does not have to be differentiable, it just needs to be continuous)