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On the Nevanlinna characteristic of $f(qz)$ and its applications. (English) Zbl 1198.30033
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 $q\in \Bbb C$ 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.

30D35Distribution of values (one complex variable); Nevanlinna theory
Full Text: DOI
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