Models for the Eremenko-Lyubich class
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909447742291968 |
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| author | Bishop, Christopher J. |
| author_facet | Bishop, Christopher J. |
| contents | If $f$ is in the Eremenko-Lyubich class (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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01894 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Models for the Eremenko-Lyubich class Bishop, Christopher J. Complex Variables 30D15 (30C62 37F10) If $f$ is in the Eremenko-Lyubich class (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 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. |
| title | Models for the Eremenko-Lyubich class |
| topic | Complex Variables 30D15 (30C62 37F10) |
| url | https://arxiv.org/abs/2501.01894 |