# Schur parameters and Carath\'eodory class

## TL;DR

This paper explores the relationship between Schur parameters and Carathéodory functions, providing a recursive formula to express coefficients and analyzing the mapping properties of these correspondences.

## Contribution

It introduces a recursive formula linking Carathéodory function coefficients to Schur parameters and studies the mapping properties of these relationships.

## Key findings

- Derived a recursive formula for Carathéodory coefficients in terms of Schur parameters.
- Established the parametrization of Carathéodory functions using independent variables.
- Analyzed the mapping properties of the correspondence between parameters and functions.

## Abstract

The Schur (resp. Carath\'eodory) 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 Carath\'eodory 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.