Extension of the theorems of Carathéodory-Toeplitz-Schur and Pick (Q1094552)

Let \(z_ 1,...,z_ k\) be distinct points in the unit disc D and let \(c_{i\alpha}\) \((\alpha =0,...,n_ i-1)\) be \(n_ i\) complex numbers for each \(z_ i\). There exists a holomorphic function f in D with \(| f| \leq 1\) whose first \(n_ i\) Taylor coefficients at \(z_ i\) are the prescribed values \(c_{i\alpha}\) if and only if the Hermitian matrix defined by \(z_ i\) and \(c_{i\alpha}\) is positive semidefinite. This result contains both Schur's theorem (1917), derived from the results of Carathéodory and Toeplitz, and Pick's theorem (1915).