On the Nevanlinna characteristic of \(f(qz)\) and its applications (Q984704)

The authors investigate the relation between the Nevanlinna characteristic functions \(T\big(r,f(qz)\big)\) and \(T\big(r,f(z)\big)\) for a zero-order meromorphic function \(f\) and a non-zero constant \(q\). It is shown that \(T\big(r,f(qz)\big)=\big(1+o(1)\big)T\big(r,f(z)\big)\) for all \(r\) in a set of lower logarithmic density 1. This estimate is sharp in the sense that, for any \(q\in \mathbb C\) such that \(|q|\neq 1\), and all \(\rho >0\), there exists a meromorphic function \(h\) of order \(\rho \) such that \(T\big(r,h(qz)\big)=\big(|q|^\rho +o(1)\big)T\big(r,h(z)\big)\) as \(r\rightarrow \infty \) outside of an exceptional set of finite linear measure. As applications, they 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.