Let $u$ be a subharmonic function on $\mathbb{C}$. Suppose that $$\limsup_{z\to \infty} \frac{u(z)}{\log|z|}=0$$
I'm trying to prove that this implies $u(z)$ is constant. I have a feeling that it may have to do with Hadamard's Three Circles Theorem and/or the maximum principle for sub/superharmonic functions, but I'm not getting anywhere.