An application from partial sums of \(e^ z\) to a problem in several complex variables
type: Item
instance of: scholarly article
title: An application from partial sums of \(e^ z\) to a problem in several complex variables
zbMATH Open document ID: 0802.32010
DOI: 10.1016/0377-0427(93)90301-Q
author: Roger W. Barnard
author: Kent Pearce
author: Richard S. Varga
published in: Journal of Computational and Applied Mathematics
publication date: 1994-12-12T00:00:00Z
review text: \textit{I. Graham} has conjectured in Complex Variables, Theory Appl. 15, No. 1, 37-42 (1990; Zbl 0681.32002) the following \(n\)-dimensional version of the Koebe 1/4 theorem: Given an \(m\)-degree polynomial mapping \(f\) of \(\mathbb{C}^ n\) into \(\mathbb{C}^ n\), there exists a positive constant \(a_ m\) such that \(Hf(B_ 1)\supset B_{a_ m}\), where \(B_ r\) is the ball with radius \(r\) and \(HX\) is the closed convex hull of \(X\). In this paper it is proved that this conjecture holds.
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 32A05
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 219126
zbMATH Keywords: Koebe 1/4 theorem
zbMATH Keywords: polynomial mapping
MaRDI profile type: Publication
cites work: The growth and 1/4-theorems for starlike mappings in \(\mathbb{C}^ n\)
cites work: A distortion theorem for biholomorphic mappings in \({\mathbb{C}}^ 2\)
cites work: A Characterization of the Exponential Series
cites work: Asymptotics for the zeros of the partial sums of \(e^ z\). I
cites work: Q3660033
cites work: Distortion in several variables
cites work: Some bounds on convex mappings in several complex variables
cites work: Distortion theorems for holomorphic maps between convex domains in n
cites work: Some Remarks on the Intrinsic Measures of Eisenman
cites work: Q5827033
cites work: The Strong Maximum Modulus Theorem for Analytic Functions into a Banach Space
cites work: Q3998905
cites work: Q3361083
full work available at URL: 0377-0427(93)90301-q
OpenAlex ID: W2064657871
description: scientific article; zbMATH DE number 219126