Summary: M. J. Ablowitz, R. Halburd and B. Herbst [Nonlinearity 13, No. 3, 889-905 (2000; Zbl 0956.39003)] applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. A typical example of their results tells us that if a complex difference equation
with rational in both arguments admits a transcendental meromorphic solution of finite order, then .
Improvements and extensions of such results are presented in this paper. In addition to order considerations, a result is proved to indicate that solutions having Borel exceptional zeros and poles seem to appear in special situations only.