# A topological dichotomy with applications to complex analysis

Colloquium Mathematicae (2015)

- Volume: 139, Issue: 1, page 137-146
- ISSN: 0010-1354

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topIosif 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 -

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