  This paper consists of three parts. First, we give so far the best condition
under which the shift invariance of the counting function, and of the
characteristic of a subharmonic function, holds. Second, a difference analogue
of logarithmic derivative of a $\delta$-subharmonic function is established
allowing the case of hyper-order equal to one and minimal hyper-type, which
improves the condition of the hyper-order less than one. Finally, we make a
careful discussion of a well-known difference equation and give out the
possible forms of the equation under a growth condition for the solutions.
