Extension of the theorems of Carathéodory-Toeplitz-Schur and Pick

Sechiko Takahashi

Vol. 138 (1989), No. 2, 391–399

DOI: 10.2140/pjm.1989.138.391

Abstract

Let z₁,…,z_k be distinct points in the unit disc D and let c_iα (α = 0,…,n_i − 1) be n_i complex numbers for each z_i. There exists a holomorphic function f in D with |f|≤ 1 whose first n_i Taylor coefficients at z_i are the prescribed values c_iα if and only if the Hermitian matrix defined by z_i and c_iα is positive semidefinite.

Mathematical Subject Classification 2000

Primary: 30E05

Milestones

Received: 28 January 1988

Published: 1 June 1989