A topological dichotomy with applications to complex analysis

Iosif Pinelis. "A topological dichotomy with applications to complex analysis." Colloquium Mathematicae 139.1 (2015): 137-146. <http://eudml.org/doc/283973>.

@article{IosifPinelis2015,
abstract = { 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. },
author = {Iosif Pinelis},
journal = {Colloquium Mathematicae},
keywords = {compact topological spaces; continuous maps, holomorphic maps, conformal maps; quasiconformal maps},
language = {eng},
number = {1},
pages = {137-146},
title = {A topological dichotomy with applications to complex analysis},
url = {http://eudml.org/doc/283973},
volume = {139},
year = {2015},
}

TY - JOUR
AU - Iosif Pinelis
TI - A topological dichotomy with applications to complex analysis
JO - Colloquium Mathematicae
PY - 2015
VL - 139
IS - 1
SP - 137
EP - 146
AB - 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.
LA - eng
KW - compact topological spaces; continuous maps, holomorphic maps, conformal maps; quasiconformal maps
UR - http://eudml.org/doc/283973
ER -