A topological dichotomy with applications to complex analysis

Iosif Pinelis¹

¹ Department of Mathematical Sciences Michigan Technological University Houghton, MI 49931, U.S.A.

Colloquium Mathematicum, Tome 139 (2015) no. 1, pp. 137-146

Résumé

Let $X$ be a compact topological space, and let $D$ be a subset of $X$. Let $Y$ be a Hausdorff topological space. Let $f$ be a continuous map of the closure of $D$ to $Y$ such that $f(D)$ is open. Let $E$ be any connected subset of the complement (to $Y$) of the image $f(\partial D)$ of the boundary $\partial D$ of $D$. Then $f(D)$ either contains $E$ or is contained in the complement of $E$. Applications of this dichotomy principle are given, in particular for holomorphic maps, including maximum and minimum modulus principles, an inverse boundary correspondence, and a proof of Haagerup's inequality for the absolute power moments of linear combinations of independent Rademacher random variables. (A three-line proof of the main theorem of algebra is also given.) More generally, the dichotomy principle is naturally applicable to conformal and quasiconformal mappings.

Détail

DOI : 10.4064/cm139-1-9

Keywords: compact topological space subset hausdorff topological space continuous map closure connected subset complement image partial boundary partial either contains contained complement applications dichotomy principle given particular holomorphic maps including maximum minimum modulus principles inverse boundary correspondence proof haagerups inequality absolute power moments linear combinations independent rademacher random variables three line proof main theorem algebra given generally dichotomy principle naturally applicable conformal quasiconformal mappings