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