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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2510.14736 |
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| _version_ | 1866909850225606656 |
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| author | Wu, Shijian |
| author_facet | Wu, Shijian |
| contents | In this paper, we study delay differential equations involving the Schwarzian derivative $S(f,z)$, expressed in the form
\begin{equation*}
f(z+1)f(z-1) + a(z)S(f,z) =R(z,f(z))= \frac{P(z,f(z))}{Q(z,f(z))}
\end{equation*}
where $a(z)$ is rational, $P(z,f)$ and $Q(z,f)$ are coprime polynomials in $f$ with rational coefficients.
Our main result shows that if a subnormal transcendental meromorphic solution exists, then the rational function $R(z,f)=P(z,f)/Q(z,f)$ satisfies $°_fR\leq 7$ and $°_fP\leq °_fQ +2$, where $°_fR =\max\{°_fP, °_fQ\}.$
Furthermore, for any rational root $b_1$ of $Q(z,f)$ in $f$ with multiplicity $k$, we show that $k \leq 2$.
Finally, a classification of such equations is provided according to the multiplicity structure of the roots of $Q(z,f)$. Some examples are given to support these results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Admissible solutions of delay Schwarzian differential equations Wu, Shijian Complex Variables In this paper, we study delay differential equations involving the Schwarzian derivative $S(f,z)$, expressed in the form \begin{equation*} f(z+1)f(z-1) + a(z)S(f,z) =R(z,f(z))= \frac{P(z,f(z))}{Q(z,f(z))} \end{equation*} where $a(z)$ is rational, $P(z,f)$ and $Q(z,f)$ are coprime polynomials in $f$ with rational coefficients. Our main result shows that if a subnormal transcendental meromorphic solution exists, then the rational function $R(z,f)=P(z,f)/Q(z,f)$ satisfies $°_fR\leq 7$ and $°_fP\leq °_fQ +2$, where $°_fR =\max\{°_fP, °_fQ\}.$ Furthermore, for any rational root $b_1$ of $Q(z,f)$ in $f$ with multiplicity $k$, we show that $k \leq 2$. Finally, a classification of such equations is provided according to the multiplicity structure of the roots of $Q(z,f)$. Some examples are given to support these results. |
| title | Admissible solutions of delay Schwarzian differential equations |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2510.14736 |