A topological dichotomy with applications to complex analysis
Document Type
Article
Publication Date
2015
Department
Department of Mathematical Sciences
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.
Publication Title
Colloquium Mathematicum
Recommended Citation
Pinelis, I.
(2015).
A topological dichotomy with applications to complex analysis.
Colloquium Mathematicum,
139(1), 137-146.
http://doi.org/10.4064/cm139-1-9
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/14218
