A topological dichotomy with applications to complex analysis

• Autorzy: Iosif Pinelis
• Języki publikacji: EN

Abstrakty

EN

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.

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

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 139
• Numer: 1
• Strony: 137-146

Daty

wydano

2015

Twórcy

autor

Iosif Pinelis

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

Identyfikatory

DOI

10.4064/cm139-1-9