Finite boundary interpolation by univalent functions
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 \(k=1,2,...,n\).
scientific article; zbMATH DE number 4082185