The Szegő kernel and proper holomorphic mappings to a half plane
type: Item
instance of: scholarly article
title: The Szegő kernel and proper holomorphic mappings to a half plane
zbMATH Open document ID: 1234.30008
DOI: 10.1007/BF03321796
author: Steven R. Bell
published in: Computational Methods and Function Theory
publication date: 2011-10-28T00:00:00Z
full work available at URL: cmf11011.htm
review text: The author uses the Szegő kernel \(S(z,w)\) and the Garabedian kernel \(L(z,w)\) to describe the proper holomorphic mappings from a bounded, \(n\)-connected planar domain (whose \(n\) boundary components are smooth, real-analytic Jordan curves) onto the right-hand half-plane. One of the main results states that if \(b_1\), \(\ldots\), \(b_n\) are arbitrary points in the boundary of the domain (one point from each boundary curve), then there exist another boundary point \(a\) and positive real numbers \(c_1\), \(\ldots\), \(c_n\) such that \[ \sum_{j=1}^n c_j \frac{S(z,b_j) L(b_j,a)} {L(z,a)} \] is an \(n\)-to-\(1\) proper holomorphic mapping to the right-hand half-plane having poles at the points \(b_{1}\), \(\ldots\), \(b_{n}\). Moreover, these particular mappings generate the additive semigroup of all proper holomorphic mappings to the right-hand half-plane. An application is an easy construction of a primitive pair for the space of meromorphic functions on the double of a finitely connected planar domain.
reviewed by: Harold P. Boas
Mathematics Subject Classification ID: 30C35
zbMATH DE Number: 5965034
zbMATH Keywords: Ahlfors map
zbMATH Keywords: Garabedian kernel
zbMATH Keywords: multiply connected domains
zbMATH Keywords: primitive pairs
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5965034