Admissible Solutions of the Schwarzian Type Difference Equation

Abstract

This paper is to investigate the Schwarzian type difference equation ${\left[\left({\Delta }^{3}f/\Delta f\right)-\left(3/2\right){\left({\Delta }^{2}f/\Delta f\right)}^2\right]}^k=R\left(z,f\right)=\left(P(z,f)/Q(z,f)\right),$ where $R(z,f)$ is a rational function in $f$ with polynomial coefficients, $P(z,f)$, respectively $Q(z,f)$ are two irreducible polynomials in $f$ of degree $p$, respectively $q$. Relationship between $p$ and $q$ is studied for some special case. Denote $d=\max \left\{p,q\right\}$. Let $f(z)$ be an admissible solution of $(*)$ such that ; then for $s$ (≥2) distinct complex constants ${\alpha }_1,\dots ,{\alpha }_s$, $q+2k{\sum }_{j=1}^s\delta ({\alpha }_j,f)\le 8k.$ In particular, if $N(r,f)=S(r,f)$, then $d+2k{\sum }_{j=1}^s\delta ({\alpha }_j,f)\le 4k.$