Cpx Analysis: Harmonic Function u(z) + ln|z| >= 0

Denote the open unit disk as $\displaystyle \mathbb{D}$. Let $\displaystyle u(z)$ be harmonic and positive for $\displaystyle z \in \mathbb{D} \backslash \{ 0 \}$. Suppose that $\displaystyle u(z) + ln|z|$ is harmonic for $\displaystyle z \in \mathbb{D}$.

Claim: $\displaystyle \forall \ z \in \mathbb{D}, u(z) + ln|z| \ge 0$.

This is an old prelim problem I'm trying to figure out. Looks like it could be an application of the max. principle but I'm not seeing the way through.