LAUSR

Let the function $$\varphi $$φ be holomorphic in the unit disk $$\mathbb {D}$$D and let $$\varphi (\mathbb {D})\subset \mathbb {D}$$φ(D)⊂D. We consider points $$\zeta \in \partial \mathbb {D}$$ζ∈∂D where $$\varphi $$φ has an angular limit $$\varphi (\zeta )\in \partial \mathbb {D}$$φ(ζ)∈∂D and study the behaviour of $$(\varphi (z)-\varphi (\zeta ))/(z-\zeta )$$(φ(z)-φ(ζ))/(z-ζ) as z tends to $$\zeta $$ζ in various ways. In particular, there is a result connecting $$|\varphi '(\zeta _{\nu })|$$|φ′(ζν)| and $$|\varphi (\zeta _{\mu })-\varphi (\zeta _{\nu })|$$|φ(ζμ)-φ(ζν)| for three points $$\zeta _{\nu }$$ζν. Expressed as a positive semidefinite quadratic form, this result could, perhaps, be generalized to n points $$\zeta _{\nu }$$ζν.