= One-sided bounded harmonic functions on the punctured plane are constant
A <harmonic function> on $\mathbb R^2\setminus\{0\}$ bounded on one side is constant. Shift and change sign to make it nonnegative. Its circular mean has the form $a\log r+b$; positivity for every $r>0$ forces $a=0$. For $|x|=r$, the <mean value property for harmonic functions> bounds its value by the mean integral over $r/2<|z|<3r/2$, giving at most $8b$. A <removable singularity for a bounded harmonic function> fills the origin, and the <harmonic Liouville theorem> finishes the proof. The dimensional distinction matters: $|x|^{2-n}$ is a positive nonconstant <harmonic function> on punctured $\mathbb R^n$ for $n\geq3$.
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