Subnormal transcendental meromorphic solutions of difference equations with Schwarzian derivative

Abstract

The existence of subnormal solutions of following three difference equations with Schwarzian derivative $$\ddot{I}‰(z+1)-\ddot{I}‰(z-1)+a(z)(S(\ddot{I}‰,z){)}^n=R(z,\ddot{I}‰(z)),$$ $$\ddot{I}‰(z+1)\ddot{I}‰(z-1)+a(z)S(\ddot{I}‰,z)=R(z,\ddot{I}‰(z)),$$ and $$(\ddot{I}‰(z)\ddot{I}‰(z+1)-1)(\ddot{I}‰(z)\ddot{I}‰(z-1)-1)+a(z)S(\ddot{I}‰,z)=R(z,\ddot{I}‰(z))$$ are studied by using Nevanlinna theory, where $n\ge 1$ is an integer, $a(z)$ is small with respect to $\ddot{I}‰$, $S(\ddot{I}‰,z)$ is Schwarzian derivative, $R(z,\ddot{I}‰)$ is rational in $\ddot{I}‰$ with small meromorphic coefficients with respect to $\ddot{I}‰$. 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.