Finite boundary interpolation by univalent functions
type: Item
instance of: scholarly article
title: Finite boundary interpolation by univalent functions
zbMATH Open document ID: 0662.41002
DOI: 10.1016/0021-9045(88)90045-7
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: The authors concerned with functions which are analytic and univalent in \(D=\{z\in {\mathbb{C}}:| z| <1\}\) and satisfy the boundary interpolation \(f(z_ k)=w_ k\) for \(k=1,2,...,n\). The following main theorems are proved: Theorem 1. There is a function f which is analytic and univalent in the union of D with a neighborhood of \(\{z_ 1,z_ 2,...,z_ n\}\) and continuous on \(\bar D\) such that \(f(z_ k)=w_ k\) for \(k=1,2,...,n\). Furthermore, \(| f(z)| =1\) if \(| z| =1\) and z is sufficiently near to any of the points \(z_ k\). Theorem 2. Suppose \(a<x_ 1<x_ 2<...<x_ n<b\) and \(y_ 1<y_ 2<...<y_ n\). There is a real-valued polynomial p which is univalent in a domain containing [a,b] such that \(p(x_ k)=y_ k\) for \(k=1,2,...,n\).
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 4082185
zbMATH Keywords: analytic function
zbMATH Keywords: real-valued polynomial
author: Thomas H. MacGregor
author: David E. Tepper
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90045-7
OpenAlex ID: W2080989024
cites work: Convex Hulls of Some Classical Families of Univalent Functions
cites work: An Elementary Interpolation Theorem
cites work: A peaking and interpolation problem for univalent functions
cites work: Q3848301
cites work: Successive Coefficients of Univalent Functions
description: scientific article; zbMATH DE number 4082185