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On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. (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 $|f\left(z+\eta \right)/f\left(z\right)|\le exp\left(|z{|}^{\sigma -1+\epsilon }\right)$ 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

$\frac{f\left(z+\eta \right)}{f\left(z\right)}=exp\left(\eta \frac{{f}^{\text{'}}\left(z\right)}{f\left(z\right)}+O\left({r}^{\beta +\epsilon }\right)\right)$

for $|z|$ 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 $\beta =max\left\{\sigma -2,2\lambda -2\right\}$ if $\lambda <1$ and $\beta =max\left\{\sigma -2,\lambda -1\right\}$ if $\lambda \ge 1$.

The above result is used to show that

$\frac{f\left(z+\eta \right)-f\left(z\right)}{f\left(z\right)}=\eta \frac{{f}^{\text{'}}\left(z\right)}{f\left(z\right)}+O\left({r}^{2\sigma -2+\epsilon }\right)$

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.

##### MSC:
 30D30 General theory of meromorphic functions 30D35 Distribution of values (one complex variable); Nevanlinna theory 39A05 General theory of difference equations 46E25 Rings and algebras of continuous, differentiable or analytic functions