I have a set $\Omega$, a proper subset of the extended complex plane, that is open, connected, and contains $\infty$. h is a harmonic function on this set and it has a removable singularity at $\infty$. In addition, h is bounded outside neighborhoods of $\infty$, which (I think) combined with the removable singularity result means it's bounded on all of $\Omega$. Is h necessarily constant?
My instinct is that the limit at $\infty$ (let's call it M) is a bound for the function, and therefore h has a local extremum at $\infty$. Then the min/max principle for harmonic functions would say that h is constant on $\Omega$. It seems really obvious to me that h must be monotonic as z approaches $\infty$, but I can't quite say why, which makes me worry I'm missing something. There are more details that I used earlier in the problem, but I don't think they're necessary for this part.
$\Omega=\{z\in \mathbb C: |z| >1\}\cup \{\infty\}$. $f(z)=\Re (\frac 1 z)$ is a counter-example.