The equality case in the Schwarz-Pick lemma at the boundary

Denote 𝔻:= {𝑧 ∈ ℂ : |𝑧| < 1}. The classical result of Schwarz-Pick states that every holomorphic function 𝑓 : 𝔻→ 𝔻 is a contraction with respect to the hyperbolic distance 𝑑_𝔻. Equivalently, 𝑓 satisfies the (two-point) Schwarz-Pick inequality

𝑑_𝔻 (𝑓(𝑧), 𝑓(w))
≤ 𝑑_𝔻(𝑧, w),         𝑧, w ∈ 𝔻.

There also is a (one-point) version of Schwarz-Pick inequality in terms of the hyperbolic derivative 𝐷_ℎ𝑓, that is

𝐷_ℎ𝑓(𝑧) := lim_𝑤→𝑧𝑑_𝔻 (𝑓(𝑧), 𝑓(𝑤))/𝑑_𝔻(𝑧, 𝑤) ≤ 1,         𝑧 ∈ 𝔻.

The case of equality in the above inequalities can be used to characterize specific functions: equality holds at two different points resp. one point if and only if 𝑓 is a holomorphic automorphism of 𝔻. 

This talk deals with equality at the boundary in the Schwarz-Pick inequality. Here, we distinguish two types of “boundary Schwarz-Pick lemmas”: On the one hand, we discuss different ways to have equality at a boundary point σ ∈ ∂𝔻 in terms of 𝑑_𝔻 and 𝐷_ℎ𝑓 in an appropriate asymptotic sense that determine the boundary behaviour of a given holomorphic function 𝑓 : 𝔻→ 𝔻. This way we obtain boundary Schwarz-Pick (in-)equalities that characterize 𝑓 having finite angular derivative at σ or satisfying a weaker property of angle-preservation at σ. On the other hand, sufficiently strong control of the asymptotic behaviour of 𝑓 resp. 𝐷_ℎ𝑓 at σ allows for rigidity results that are more in analogy to the original spirit of Schwarz-Pick lemma in the sense that they characterize specific functions.