"Find all harmonic functions $f:\mathbb{C}\backslash\{0\} \to \mathbb{R}$ that are constant on every circle centered at 0." This is one of the past qualifying exam problems that I was working on.
I was thinking to deal with $\frac{1}{f}$ so that $\frac{1}{f}$ is defined at 0 and use Schwarz lemma or something like that.
Any help or guidance would be really appreciated.
Thank you in advance.