Subnormal transcendental meromorphic solutions of difference equations with Schwarzian derivative
M. T. Xia, J. R. Long, X. X. Xiang
TL;DR
The article advances the study of subnormal transcendental meromorphic solutions for three delay-difference equations with Schwarzian derivatives, deriving sharp necessary conditions on the degree of the rational nonlinearity $R(z,\omega)$ and on the polynomials $P$, $Q$ when expressed as $R(z,\omega)=P(z,\omega)/Q(z,\omega)$. Employing Nevanlinna theory, Valiron–Mohon'ko identities, and differential-difference lemmas for the Schwarzian operator, the authors obtain explicit bounds: for the first equation $\deg_{\omega}(R)\le5n+2$ with $\deg_{\omega}(P)$ and $\deg_{\omega}(Q)$ constrained by $n$; for the Painlevé V-type equation, $\deg_{\omega}(R)\le9$ with related bounds on $\deg_{\omega}(P)$ and $\deg_{\omega}(Q)$ and a root-multiplicity cap $k\le2$; and they demonstrate sharpness and provide illustrative examples. The results extend discrete Malmquist–Yosida–Steinmetz-type classifications to Schwarzian-difference settings, linking subnormal growth to structural constraints in discrete integrable systems. This deepens understanding of when such highly nontrivial solutions can exist and highlights the interplay between growth, pole/zero distribution, and Schwarzian dynamics in difference equations.
Abstract
The existence of subnormal solutions of following three difference equations with Schwarzian derivative $$ω(z+1)-ω(z-1)+a(z)(S(ω,z))^n=R(z,ω(z)),$$ $$ω(z+1)ω(z-1)+a(z)S(ω,z)=R(z,ω(z)),$$ and $$(ω(z)ω(z+1)-1)(ω(z)ω(z-1)-1)+a(z)S(ω,z)=R(z,ω(z))$$ are studied by using Nevanlinna theory, where $n\ge 1$ is an integer, $a(z)$ is small with respect to $ω$, $S(ω,z)$ is Schwarzian derivative, $R(z,ω)$ is rational in $ω$ with small meromorphic coefficients with respect to $ω$. The necessary conditions for the existence of subnormal transcendental meromorphic solutions of the above equations are obtained. Some examples are given to support these results.