Admissible solutions of delay Schwarzian differential equations

Abstract: 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.