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Models for the Eremenko–Lyubich class
Author(s) -
Bishop Christopher J.
Publication year - 2015
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdv021
Subject(s) - holomorphic function , class (philosophy) , simple (philosophy) , property (philosophy) , mathematics , bounded function , transcendental function , set (abstract data type) , function (biology) , pure mathematics , open set , transcendental number , discrete mathematics , mathematical analysis , computer science , artificial intelligence , philosophy , epistemology , evolutionary biology , biology , programming language
If f is in the Eremenko–Lyubich class B (transcendental entire functions with bounded singular set), then Ω = { z : | f ( z ) | > R } andf | Ω must satisfy certain simple topological conditions when R is sufficiently large. A model ( Ω , F ) is an open set Ω and a holomorphic function F on Ω that satisfy these same conditions. We show that any model can be approximated by an Eremenko–Lyubich function in a precise sense. In many cases, this allows the construction of functions in B with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do.