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Abstrakty
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.
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.
Słowa kluczowe
Kategorie tematyczne
- 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
- 30G12: Finely holomorphic functions and topological function theory
- 54C05: Continuous maps
- 30A10: Inequalities in the complex domain
- 60E15: Inequalities; stochastic orderings
- 30E25: Boundary value problems
- 30C25: Covering theorems in conformal mapping theory
- 30C80: Maximum principle; Schwarz's lemma, Lindel\"of principle, analogues and generalizations; subordination
- 30C35: General theory of conformal mappings
Czasopismo
Rocznik
Tom
Numer
Strony
137-146
Opis fizyczny
Daty
wydano
2015
Twórcy
autor
- Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, U.S.A.
Bibliografia
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-cm139-1-9