*(English)*Zbl 1172.30009

For a meromorphic function $f$ of order $\sigma $, the logarithmic derivative ${f}^{\text{'}}/f$ satisfies the estimate $|{f}^{\text{'}}{\left(z\right)/f\left(z\right)|\le |z|}^{\sigma -1+\epsilon}$ 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 $|f(z+\eta )/f\left(z\right)|\le exp\left(\right|z{|}^{\sigma -1+\epsilon})$ 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 $\left|z\right|$ outside a set of finite logarithmic measure, where $\beta $ is defined as follows: denoting by $\lambda $ the maximum of the exponents of convergence of the zeros and poles of $f$, we have $\beta =max\{\sigma -2,2\lambda -2\}$ if $\lambda <1$ and $\beta =max\{\sigma -2,\lambda -1\}$ if $\lambda \ge 1$.

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.