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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 \(f'/f\) satisfies the estimate \(|f'(z)/f(z)|\leq|z|^{\sigma-1+\varepsilon}\) 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 \(|f(z+\eta)/f(z)|\leq \exp(|z|^{\sigma-1+\varepsilon})\) 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 \[ \frac{f(z+\eta)}{f(z)}=\exp\left(\eta\frac{f'(z)}{f(z)}+O(r^{\beta+\varepsilon})\right) \] 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 \(\beta=\max\{\sigma-2,2\lambda-2\}\) if \(\lambda<1\) and \(\beta=\max\{\sigma-2,\lambda-1\}\) if \(\lambda\geq 1\).
The above result is used to show that \[ \frac{f(z+\eta)-f(z)}{f(z)}=\eta \frac{f'(z)}{f(z)}+O\left(r^{2\sigma - 2+\varepsilon}\right) \] 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 Meromorphic functions of one complex variable (general theory)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A05 General theory of difference equations
46E25 Rings and algebras of continuous, differentiable or analytic functions
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