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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 Tr , f ( q z ) and Tr , f ( z ) for a zero-order meromorphic function f and a non-zero constant q. It is shown that Tr , f ( q z )=1 + o ( 1 )Tr , f ( z ) for all r in a set of lower logarithmic density 1. This estimate is sharp in the sense that, for any q such that |q|1, and all ρ>0, there exists a meromorphic function h of order ρ such that Tr , h ( q z )=|q| ρ + o (1)Tr , h ( z ) as r 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.

MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory
References:
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