Let $u$ be a harmonic function on the unit disk $\Delta$, taking values in $[0,1]$.
Is it true that this implies that $u$ is Lipschitz for the Poincaré metric ?
If not, what can be said about a harmonic function satisfying this property ?
I am thinking the answer to 1. might be true, but I'm not sure. I was thinking that since the unit disk is simply connected, we can find globally a holomorphic function of which $u$ is the real part and maybe somehow manage to apply Schwarz Pick lemma. But I'm not convinced it always possible to do this, and I would be interested in a characterization (ideally) or necessary conditions on $u$ to verify this.