Limit conditions of a subharmonic function imply that it is constant

Let $u$ be a subharmonic function on $\mathbb{C}$. Intend that $$\limsup_{z\to \infty} \frac{u(z)}{\log|z|}=0$$

I'm attempting to confirm that this indicates $u(z)$ is constant. I sense that it might concern Hadamard is Three Circles Theorem and/or the maximum concept for sub/superharmonic features, yet I'm not obtaining anywhere.

2022-07-25 20:43:44
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