On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

We investigate the growth of the Nevanlinna characteristic of f ( z + η ) for a fixed η ∈ C in this paper. In particular, we obtain a precise asymptotic relation between T ( r , f ( z + η )) and T ( r , f ), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f ( z + η )/ f ( z ) which is a discrete version of the classical logarithmic derivative estimates of f ( z ). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935 ) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000 ) concerning integrable difference equations.