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

Vladimir Bolotnikov, William & Mary

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(I) (z - t(0)) + ... + s(N) (z - t(0))(N) + o(vertical bar z - t(0)vertical bar(N)) at a given point to on the unit circle. (C) 2011 Elsevier Inc. All rights reserved.