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Bohr’s Phenomenon on a Regular Condenser in the Complex Plane

Bohr’s Phenomenon on a Regular Condenser in the Complex Plane We prove the following generalization of Bohr’s Theorem: let $K\subset{\rm C}$ be a continuum, $(F_{K,n})_{n\geq 0}$ its Faber polynomials, ωnr the level sets of the Green function of ℂ K with singularity at infinity. Then there exists a radius $R_{0}>0$ such that for any $f=\sum_{n}a_{n}F_{K,n}\in {O}(\Omega_{R_0})$ with $f(\Omega_{R_0})\subset D(0,1) {\rm \ we \ have} \sum_{n}|a_{n}|.||F_{K,n}||_K<1$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Bohr’s Phenomenon on a Regular Condenser in the Complex Plane

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Publisher
Springer Journals
Copyright
Copyright © 2012 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321811
Publisher site
See Article on Publisher Site

Abstract

We prove the following generalization of Bohr’s Theorem: let $K\subset{\rm C}$ be a continuum, $(F_{K,n})_{n\geq 0}$ its Faber polynomials, ωnr the level sets of the Green function of ℂ K with singularity at infinity. Then there exists a radius $R_{0}>0$ such that for any $f=\sum_{n}a_{n}F_{K,n}\in {O}(\Omega_{R_0})$ with $f(\Omega_{R_0})\subset D(0,1) {\rm \ we \ have} \sum_{n}|a_{n}|.||F_{K,n}||_K<1$ .

Journal

Computational Methods and Function TheorySpringer Journals

Published: Aug 24, 2011

References