标题： 延迟Schwarz微分方程的可允许解 显示英文标题

标题： Admissible solutions of delay Schwarzian differential equations

摘要： 在本文中，我们研究涉及 Schwarzian 导数$S(f,z)$的延迟微分方程，其形式为 \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*} 其中$a(z)$是有理函数，$P(z,f)$和$Q(z,f)$是在$f$中具有有理系数的互质多项式。 我们的主要结果表明，如果存在一个次正规的超越有理函数解，则有理函数$R(z,f)=P(z,f)/Q(z,f)$满足$\deg_fR\leq 7$和$\deg_fP\leq \deg_fQ +2$，其中$\deg_fR =\max\{\deg_fP, \deg_fQ\}.$。此外，对于任何有理根$b_1$在$Q(z,f)$中的$f$，其重数为$k$，我们证明了$k \leq 2$。 最后，根据$Q(z,f)$的根的重数结构对这类方程进行了分类。给出了一些例子来支持这些结果。 显示英文摘要

摘要： 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 $\deg_fR\leq 7$ and $\deg_fP\leq \deg_fQ +2$, where $\deg_fR =\max\{\deg_fP, \deg_fQ\}.$ 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.