On a ``much underestimated'' paper of Alexander
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 (useless) condition: \(f(z)\) is univalent is equivalent to \[ {f(z_2) - f(z_1) \over z_2-z_1} \not=0 \] for all \(z_1\), \(z_2\) in the disk, into a number of \textsl{useful} sufficient conditions for a function \(w=f(z)\) to map the interior \(|z|<1\) of the unit disk in a one-to-one (univalent) manner. Eduard Study (1862-1930) proved that a sufficient condition for (convex) mapping \(f\) of the unit disk onto convex domains was \[ 1+\Re \left[ {zf''(z) \over f'(z)}\right]>0, \;f'(z)\not=0. \] Alexander constructed several families of functions, together with sufficient conditions such as the total amount of angular turning of the tangent to the boundary curve, such that the image of the unit disk was a convex region, a starlike region, and other similar regions. Today the Bieberbach conjecture on the coefficients of power series which accomplish this is settled, and the field Alexander viewed is enclosed, cultivated and domesticated, using the analytical tools with which we are familiar. But Alexander gives us a reminder that there are other tools, other arguments, other points of view. The paper is worth reading, but be prepared to work.
scientific article; zbMATH DE number 1513376