Admissible Solutions of the Schwarzian Type Difference Equation

Abstract EN

This paper is to investigate the Schwarzian type difference equation Δ3f/Δf-3/2Δ2f/Δf2k=Rz,f=P(z,f)/Q(z,f), 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=maxp,q.

Let f(z) be an admissible solution of (*) such that ρ2(f)<1; then for s  (≥2) distinct complex constants α1,…,αs , q+2k∑j=1sδ(αj,f)≤ 8k.

In particular, if N(r,f)=S(r,f), then d+2k∑j=1sδ (αj,f)≤4k.