Maximum and average valence of meromorphic functions
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910307725606912 |
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| author | Hinkkanen, Aimo Miles, Joseph |
| author_facet | Hinkkanen, Aimo Miles, Joseph |
| contents | If $f$ is a meromorphic function from the complex plane ${\mathbb C}$ to the extended complex plane $\overline{ {\mathbb C} }$, for $r > 0$ let $n(r)$ be the maximum number of solutions in $\{z\colon |z| \leq r \}$ of $f(z) = a$ for $a \in \overline{ {\mathbb C} }$, and let $A(r,f)$ be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which $\liminf_{r\to\infty} n(r)/A(r,f) \geq 1.07328$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13808 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum and average valence of meromorphic functions Hinkkanen, Aimo Miles, Joseph Complex Variables 30D35 If $f$ is a meromorphic function from the complex plane ${\mathbb C}$ to the extended complex plane $\overline{ {\mathbb C} }$, for $r > 0$ let $n(r)$ be the maximum number of solutions in $\{z\colon |z| \leq r \}$ of $f(z) = a$ for $a \in \overline{ {\mathbb C} }$, and let $A(r,f)$ be the average number of such solutions. Using a technique introduced by Toppila, we exhibit a meromorphic function for which $\liminf_{r\to\infty} n(r)/A(r,f) \geq 1.07328$. |
| title | Maximum and average valence of meromorphic functions |
| topic | Complex Variables 30D35 |
| url | https://arxiv.org/abs/2401.13808 |