Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]