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(∂D) of the boundary ∂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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm139-1-9,
author = {Iosif Pinelis},
title = {A topological dichotomy with applications to complex analysis},
journal = {Colloquium Mathematicae},
volume = {139},
year = {2015},
pages = {137-146},
zbl = {1314.30067},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm139-1-9}
}
Iosif Pinelis. A topological dichotomy with applications to complex analysis. Colloquium Mathematicae, Tome 139 (2015) pp. 137-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm139-1-9/