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

Abstract

Characterization of Schur-class functions (analytic and bounded by one in modulus on the open unit disk) in terms of their Taylor coefficients at the origin is due to I. Schur. We present a boundary analog of this result: necessary and sufficient conditions are given for the existence of a Schur-class function with the prescribed nontangential boundary expansion f ( z ) = s 0 + s 1 ( z − t 0 ) + ⋯ + s N ( z − t 0 ) N + o ( | z − t 0 | N ) at a given point t 0 on the unit circle.