On a ``much underestimated'' paper of Alexander
type: Item
instance of: scholarly article
title: On a ``much underestimated'' paper of Alexander
zbMATH Open document ID: 0979.01017
author: Alan Gluchoff
author: Frederick W. Hartmann
published in: Archive for History of Exact Sciences
publication date: 2002-02-22T00:00:00Z
review text: This is an interesting, informative but not easy-to-read paper. It has seventeen unnumbered sections. The first three introduce Alexander's paper ``Functions which map the interior of the unit disk upon simple regions'', Ann. Math. (2) 17, 12-22 (1915; JFM 45.0672.02); give a short biography of J. W. H. Alexander, and discuss Alexander's contributions to algebraic topology. The last two sections comprise an appendix discussing five points raised in the body of the paper, and a bibliography of 160 items covering seven pages. There are a dozen sections discussing Alexander's paper. Biholomorphic mappings between open sets in the complex plane are the stuff of modern complex analysis and complex manifold theory. In an open set each point will be surrounded by a disk which, without loss of generality can be taken to be a unit disk. What Alexander's paper considered is the image of such a disk: it could be another disk of the shape of a picture of Marilyn Monroe! The authors go through the paper's eight main sections (of nine) giving Alexander's results and their later developments, giving the flavour of Alexander's unique topological arguments, under the following headings: the univalent mapping paper, and zeros in a half-plane, \S\S 1-3 of Alexander's paper: the maximal region where every polynomial of degree \(n\) (with roots in the region) is univalent; starlike and convex polynomials (including the effect of critical points such as crossovers and cusps on the boundary curve), \S 4; the maximal region for these polynomials, along with Alexander's conjecture and Kakeya's theorem, \S 5; convolutions; the bounded-turning class of functions, and the Alexander-Noshiro-Kobori theorem, \S 6; critical points on the boundary of the unit disk, \S 7; and finally close-to-convex functions and sections of power series (a finite number of initial terms of the series), \S 8. James Waddell Alexander II (1888-1971) was a practical man (mountain-climber). He replaced the known (useles…
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 01A70
Mathematics Subject Classification ID: 30-03
Mathematics Subject Classification ID: 57-03
zbMATH DE Number: 1513376
zbMATH Keywords: univalent functions
zbMATH Keywords: unit disk
zbMATH Keywords: convex domains
zbMATH Keywords: starlike region
zbMATH Keywords: JFM 45.0672.02
MaRDI profile type: Publication
full work available at URL: s004070000019
OpenAlex ID: W2074019485
DOI: 10.1007/S004070000019
description: scientific article; zbMATH DE number 1513376