Necessary and sufficient conditions for extending a function to a Carathéodory function

Abstract

A criterion deciding whether a function given by its values (with multiplicities) at a sequence of points in the disc $\mathbb D=\{|z|<1\}$ can be extended to a holomorphic function with nonnegative real part in $\mathbb D$ is stated and proved. In the case when this function is given by the values of its derivatives at $z=0$, this is the well-known Carathéodory criterion. It is also shown that Carathéodory's criterion is a consequence of Schur's criterion and, conversely, Schur's criterion follows from Carathéodory's. Bibliography: 10 titles.