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Zbl 1172.30009
Chiang, Yik-Man; Feng, Shao-Ji
On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions.
(English)
[J] Trans. Am. Math. Soc. 361, No. 7, 3767-3791 (2009). ISSN 0002-9947; ISSN 1088-6850/e

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 {\it R. G. Halburd} and {\it R. J. Korhonen} [J. Math. Anal. Appl. 314, No. 2, 477--487 (2006; Zbl 1085.30026)] and by {\it Y.-M. Chiang} and {\it S.-J. Feng} [Ramanujan J. 16, No. 1, 105--129 (2008; Zbl 1152.30024)]. \par 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 $\lambda\geq 1$. \par 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. \par 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.
[Walter Bergweiler (Kiel)]
MSC 2000:
*30D30 General theory of meromorphic functions
30D35 Distribution of values (one complex variable)
39A05 General theory of difference equations
46E25 Rings and algebras of functions with smoothness properties

Keywords: difference equation; logarithmic difference; Wiman-Valiron theory

Citations: Zbl 1085.30026; Zbl 1152.30024

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