Models for the Eremenko-Lyubich class

Christopher J. Bishop

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arXiv · 2501.01894

Models for the Eremenko-Lyubich class

Christopher J. Bishop

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Abstract

If $f$ is in the Eremenko-Lyubich class (transcendental entire functions with bounded singular set) then $\Omega= \{ z: |f(z)| > R\}$ and $f|_\Omega$ must satisfy certain simple topological conditions when $R$ is sufficiently large. A model $(\Omega, F)$ is an open set $\Omega$ and a holomorphic function $F$ on $\Omega$ 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 the Eremenko-Lyubich with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do.

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BibTeX RIS

Christopher J. Bishop. 2025-01-03. Models for the Eremenko-Lyubich class. https://arxiv.org/abs/2501.01894

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