Schur Parameters and the Carathéodory Class

Abstract

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