For a meromorphic function of order , the logarithmic derivative satisfies the estimate outside a small exceptional set. This result has many applications, in particular to complex differential equations. In the study of difference equation, a similar role is played by the estimate which was obtained independently by R. G. Halburd and R. J. Korhonen [J. Math. Anal. Appl. 314, No. 2, 477–487 (2006; Zbl 1085.30026)] and by Y.-M. Chiang and S.-J. Feng [Ramanujan J. 16, No. 1, 105–129 (2008; Zbl 1152.30024)].
In the present paper the authors establish a connection between logarithmic derivatives and differences by showing that
for outside a set of finite logarithmic measure, where is defined as follows: denoting by the maximum of the exponents of convergence of the zeros and poles of , we have if and if .
The above result is used to show that
outside the exceptional set. Extensions to higher order difference quotients are also included.
Finally the paper contains a difference version of Wiman-Valiron theory which is used to show that entire solutions of first order algebraic difference equations have positive order.