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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.18649 |
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| _version_ | 1866914052117102592 |
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| author | Liu, Xiong-Feng |
| author_facet | Liu, Xiong-Feng |
| contents | In this paper, we study a Malmquist-Yosida type theorem for Schwarzian differential equations
\begin{equation}\label{1}
S(f,z)^{m} = R(z,f) = \frac{P(z,f)}{Q(z,f)},\tag{+}
\end{equation}
where $m \in \mathbb{N}^{+}$, $P(z,f)$ and $Q(z,f)$ are irreducible polynomials in $f$ with rational coefficients. If \eqref{1} admits a transcendental meromorphic solution $f$, then by a suitable M$\mathrm{\ddot{o}}$bius transformation $f \to u$, $u$ satisfies a Riccati differential equation with small meromorphic coefficients, or one of the six types of first-order differential equations (E.2)-(E.7), or $u$ satisfies one of types (E.8)-(E.14). In addition, we improve the result of Ishizaki [6, Theorem~1.1] on Schwarzian differential equations \eqref{1} with small meromorphic coefficients when $m=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Malmquist-Yosida type theorem for Schwarzian differential equations Liu, Xiong-Feng Complex Variables In this paper, we study a Malmquist-Yosida type theorem for Schwarzian differential equations \begin{equation}\label{1} S(f,z)^{m} = R(z,f) = \frac{P(z,f)}{Q(z,f)},\tag{+} \end{equation} where $m \in \mathbb{N}^{+}$, $P(z,f)$ and $Q(z,f)$ are irreducible polynomials in $f$ with rational coefficients. If \eqref{1} admits a transcendental meromorphic solution $f$, then by a suitable M$\mathrm{\ddot{o}}$bius transformation $f \to u$, $u$ satisfies a Riccati differential equation with small meromorphic coefficients, or one of the six types of first-order differential equations (E.2)-(E.7), or $u$ satisfies one of types (E.8)-(E.14). In addition, we improve the result of Ishizaki [6, Theorem~1.1] on Schwarzian differential equations \eqref{1} with small meromorphic coefficients when $m=1$. |
| title | A Malmquist-Yosida type theorem for Schwarzian differential equations |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2509.18649 |