This might be standard or trivial, but I am unable to figure it out. Let $u$ be a harmonic function on $\mathbb{C}\setminus \{0\}=\mathbb{R}^2\setminus \{0\}$, such that $r^\delta|u|$ tends to zero as $r\rightarrow \infty$, where $r$ is the radial distance for some $\delta\in (0,1)$. Is it true that $u$ is of the form $$u(z) = a + b\log |z|$$ for some $a,b\in \mathbb{R}$?
If $u$ is positive, then by Bocher's theorem, there is a $b\in \mathbb{R}$, such that $u(z) - b\log{|z|}$ extends to a harmonic function on all of $\mathbb{R}^2$, and then by analyticity of harmonic functions (and a Liouville type argument) the difference will be a constant. Is there an argument that will work for any $u$? Or is there a counterexample?
On a related note, why does Bocher's theorem need positivity? Are there any simple counterexamples? And are there any generalizations?