A generalized Schwarz lemma at the boundary

Let $\phi$ be an analytic function mapping the unit disc $\mathbb{D}$ to itself. We generalize a boundary version of Schwarz's lemma proven by D. Burns and S. Krantz and provide sufficient conditions on the local behavior of $\phi$ near a finite set of boundary points that requires $\phi$ to be a finite Blaschke product. Afterwards, we supply several counterexamples to illustrate that these conditions may also be necessary.