On transcendental meromorphic solutions of Hayman's equation
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917000006074368 |
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| author | Zhang, Yueyang |
| author_facet | Zhang, Yueyang |
| contents | We present a complete description of the form of transcendental meromorphic solutions of the second order differential equation \begin{equation}\tag† w''w-w'^2+a w'w+b w^2=αw+βw'+γ, \end{equation} where $a$, $b$, $α$, $β$ and $γ$ are all rational functions. Together with the Wiman--Valiron theory, we then show that any transcendental meromorphic solution $w$ of equation $(†)$ has hyper-order $ς(w)\leq n$ for some integer $n\geq 0$. Moreover, if $w$ has finite order $σ(w)$, then $2σ(w)$ is a positive integer; if $β\equivγ\equiv0$ and $w$ has infinite order or if $γ\not\equiv0$ and $w$ has infinite order, then the hyper-order $ς(w)$ is a positive integer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_10587 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On transcendental meromorphic solutions of Hayman's equation Zhang, Yueyang Complex Variables Primary 30D35, Secondary 34M10 We present a complete description of the form of transcendental meromorphic solutions of the second order differential equation \begin{equation}\tag† w''w-w'^2+a w'w+b w^2=αw+βw'+γ, \end{equation} where $a$, $b$, $α$, $β$ and $γ$ are all rational functions. Together with the Wiman--Valiron theory, we then show that any transcendental meromorphic solution $w$ of equation $(†)$ has hyper-order $ς(w)\leq n$ for some integer $n\geq 0$. Moreover, if $w$ has finite order $σ(w)$, then $2σ(w)$ is a positive integer; if $β\equivγ\equiv0$ and $w$ has infinite order or if $γ\not\equiv0$ and $w$ has infinite order, then the hyper-order $ς(w)$ is a positive integer. |
| title | On transcendental meromorphic solutions of Hayman's equation |
| topic | Complex Variables Primary 30D35, Secondary 34M10 |
| url | https://arxiv.org/abs/2211.10587 |