On the Nevanlinna characteristic of f(qz) and its applications

In this paper, we investigate the relation of the Nevanlinna characteristic functions T(r,f(qz)) and T(r,f(z)) for a zero-order meromorphic function f and a non-zero constant q. It is shown that T(r,f(qz))=(1+o(1))T(r,f(z)) for all r on a set of lower logarithmic density 1. This estimate is sharp in the sense that for any q∈C such that |q|≠1, and ρ>0, there exists a meromorphic function h of order ρ such that T(r,h(qz))=(|q|ρ+o(1))T(r,h(z)) as r→∞ outside of an exceptional set of finite linear measure. As applications, we give some results on zero-order meromorphic solutions of q-difference equations, and on value distribution and uniqueness of certain types of q-difference polynomials.