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

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

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

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.