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

AbstractCharacterization 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)=s0+s1(z−t0)+⋯+sN(z−t0)N+o(|z−t0|N) at a given point t0 on the unit circle