Models for the Eremenko–Lyubich class

Abstract

If f is in the Eremenko–Lyubich class B (transcendental entire functions with bounded singular set), then Ω = { z : | f ( z ) | > R } and f | 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.