  Recent results of Davidson-Paulsen-Raghupathi-Singh give necessary and
sufficient conditions for the existence of a solution to the Nevanlinna-Pick
interpolation problem on the unit disk with the additional restriction that the
interpolant should have the value of its derivative at the origin equal to
zero. This concrete mild generalization of the classical problem is
prototypical of a number of other generalized Nevanlinna-Pick interpolation
problems which have appeared in the literature (for example, on a
finitely-connected planar domain or on the polydisk). We extend the results of
Davidson-Paulsen-Raghupathi-Singh to the setting where the interpolant is
allowed to be matrix-valued and elaborate further on the analogy with the
theory of Nevanlinna-Pick interpolation on a finitely-connected planar domain.
