Schur parameters and Carathéodory class

2019/02/06 by Li, Ming, Sugawa, Toshiyuki
#30C45 (Primary) #30C50 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.02000

Abstract

The Schur (resp. Carathéodory) class consists of all the analytic functions f on the unit disk with |f|≤ 1 (resp. \Re f>0 and f(0)=1). The Schur parameters γ₀,γ₁,… (|γ_j|≤ 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 γ₁,…, γ_n with |γ_j|≤ 1. The mapping properties of those correspondences are also studied.

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