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

Abstract

It is shown that the following conditions are equivalent for the generalized Schur class functions w at a boundary point t 0 ∈ 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]); t 0 -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.

On higher order boundary derivatives of an analytic self-map of the unit disk

“…We will identify S (0) (t 0 ) with S. The higher order Carathéodory-Julia condition (2.3) was introduced in [11] and studied later in [14,13]. This condition can be equivalently reformulated in terms of the de Branges-Rovnyak space H( f ) (we refer to [17] for the definition) associated with the function f ∈ S as follows: a Schur-class function f belongs to S (n) (t 0 ) if and only if for every h ∈ H( f ), the boundary limits h j (t 0 ) exist for j = 0, .…”

“…Theorem 3.1 states in particular that positivity of this structured matrix is an exclusive property of S (n) (t 0 )-class functions. The following stronger version of the implication (3) ⇒ (1) in Theorem 3.1 appears in Theorem 1.7 [14].…”

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

“…Equivalences (1)⇐⇒(4)⇐⇒(5), implication (5)=⇒(6) and statements (a) and (b) were proved in [10]; implication (6)=⇒(1) and equivalence (1)=⇒ (7) appear in [13] and [12], respectively. Equivalence (1)⇐⇒ (7) was established in [1] for inner and extended in [22] to general Schur class functions.…”

On Zeros of Certain Analytic Functions

“…do not have common zeros in D. It follows from the Carathéodory-Julia theorem that if a generalized Schur function g admits a unimodular boundary limit g(t 0 ), then the limit g (t 0 ) exists and equals either a real number or +∞ (see e.g., [10]). …”