When a meromorphic function that omits three values has a bounded type
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915921448140800 |
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| author | Eremenko, Alexandre Kulikov, Aleksei Sodin, Mikhail |
| author_facet | Eremenko, Alexandre Kulikov, Aleksei Sodin, Mikhail |
| contents | Suppose that a function $F$ is meromorphic in the domain
$\mathbb H(-m) = \{ z : \mathrm{Im}\, z > -m(\mathrm{Re}\, z) \}$,
where $m$ is an even, positive, and continuous function that does not increase on
$\mathbb R_{\ge 0}$, and suppose that $F$ omits there three distinct values.
Then $F$ is of bounded type in the upper half-plane
(i.e., is represented there as a quotient of two bounded analytic functions), provided that the logarithmic integral of the function $m$ is convergent. On the other hand, if the logarithmic integral of $m$ diverges, there exists a function $F$ meromorphic in $\mathbb H(-m)$, that omits there three distinct values,
and which is of unbounded type in the upper half-plane.
This result is motivated by a century old question originating with Rolf Nevanlinna, which remains unsettled. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06136 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | When a meromorphic function that omits three values has a bounded type Eremenko, Alexandre Kulikov, Aleksei Sodin, Mikhail Complex Variables 30D30 Suppose that a function $F$ is meromorphic in the domain $\mathbb H(-m) = \{ z : \mathrm{Im}\, z > -m(\mathrm{Re}\, z) \}$, where $m$ is an even, positive, and continuous function that does not increase on $\mathbb R_{\ge 0}$, and suppose that $F$ omits there three distinct values. Then $F$ is of bounded type in the upper half-plane (i.e., is represented there as a quotient of two bounded analytic functions), provided that the logarithmic integral of the function $m$ is convergent. On the other hand, if the logarithmic integral of $m$ diverges, there exists a function $F$ meromorphic in $\mathbb H(-m)$, that omits there three distinct values, and which is of unbounded type in the upper half-plane. This result is motivated by a century old question originating with Rolf Nevanlinna, which remains unsettled. |
| title | When a meromorphic function that omits three values has a bounded type |
| topic | Complex Variables 30D30 |
| url | https://arxiv.org/abs/2604.06136 |