We prove two boundary Schwarz lemmas for disks. The first one concerns holomorphic maps from the unit disk into the unit ball of \(\mathbb {C}^m\) . If \(F:\mathbb {D}\rightarrow \mathbb {B}_m\) is holomorphic, \(F(\zeta )\in \partial \mathbb {B}_m\) for some \(\zeta \in \mathbb {T}\) , and a finite angular derivative exists at that point, then \(\left\| F'(\zeta )\right\| \) admits an explicit lower bound in terms of \(F(0)\) and \(F'(0)\) . The estimate is sharp, and we identify all equality maps; in particular, the extremals are one-dimensional Blaschke-type disks. The second result is a boundary Schwarz lemma for conformal minimal disks \(F:\mathbb {D}\rightarrow \mathbb {B}^n\subset \mathbb {R}^n\) , \(n\ge 3\) . It follows from the distance-decreasing theorem of Forstnerič and Kalaj for the Poincare metric on the disk and the Cayley–Klein metric on the ball. We also determine the equality case: equality forces the image to be a totally geodesic planar disk, and in the noncentered case the boundary point is aligned with the radial direction of the center.
