An Inductive Julia-carathéodory Theorem for Pick Functions in Two Variables

We study the asymptotic behavior of Pick functions, analytic functions which take the upper half plane to itself. We show that if a two variable Pick function f has real residues to order 2N − 1 at infinity and the imaginary part of the remainder between f and this expansion is of order 2N + 1, then f has real residues to order 2N and directional residues to order 2N + 1. Furthermore, f has real residues to order 2N + 1 if and only if the 2N + 1-th derivative is given by a polynomial, thus obtaining a two variable analogue of a higher order Julia-Carathéodory type theorem.