On the Nevanlinna characteristic of and its applications

In this paper, we investigate the relation of the Nevanlinna characteristic functions T ( r , f ( q z ) ) 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 ( q z ) ) = ( 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 ( q z ) ) = ( | 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.