Carathéodory-Julia type conditions and sym- metries 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 w at a boundary point t0 ∈ T: Carathéodory– Julia type condition of order n ([2], [3]); agreeing of asymptotics from inside and outside of the disk D up to order 2n+1 ([11]); t0-isometry of the coefficients of the boundary asymptotics; a certain structured matrix P constructed from these coefficients being Hermitian ([1]). Some interconnections between these properties are established for more general classes of functions.