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.