Schur parameters and Carathéodory class

Abstract

The Schur (resp. Carathéodory) class consists of all the analytic functions $f$ on the unit disk with $|f|\le 1$ (resp. $\Re f>0$ and $f(0)=1$). The Schur parameters ${\gamma }_0,{\gamma }_1,\dots (|{\gamma }_j|\le 1)$ are known to parametrize the coefficients of functions in the Schur class. By employing a recursive formula for it, we describe the $n$-th coefficient of a Carathéodory function in terms of $n$ independent variables ${\gamma }_1,\dots ,{\gamma }_n$ with $|{\gamma }_j|\le 1.$ The mapping properties of those correspondences are also studied.

12 pages