Carathéodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk

It is shown that the following conditions are equivalent for the generalized Schur class functions at a boundary point t 0 ∈ : 1) Carathéodory–Julia type condition of order n ; 2) agreeing of asymptotics of the original function from inside and of its continuation by reflection from outside of the unit disk up to order 2 n + 1; 3) t 0 ‐isometry of the coefficients ofthe boundary asymptotics; 4) a certain structured matrix ℙ constructed from these coefficients being Hermitian. It is also shown that for an arbitrary analytic function, properties 2), 3), 4) are still equivalent to each other and imply 1) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)