Subnormal transcendental meromorphic solutions of difference equations with Schwarzian derivative

Abstract

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