Advanced Search

Query:

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

Format:

Display: entries per page entries

Zbl 1144.30012
Chen, Zong-Xuan; Shon, Kwang Ho
On zeros and fixed points of differences of meromorphic functions. (English)
[J] J. Math. Anal. Appl. 344, No. 1, 373-383 (2008). ISSN 0022-247X

This article is devoted to investigating zeros and fixed points of differences of entire and meromorphic functions in the complex plane, extending previous results due to Bergweiler and Langley [{\it W. Bergweiler} and {\it J. K. Langley}, Math. Proc. Camb. Philos. Soc. 142, No. 1, 133--147 (2007; Zbl 1114.30028)]. Denoting, for a transcendental meromorphic function $f$, $\Delta f(z):=f(z+c)-f(z)$, $\Delta^{n+1}f(z):=\Delta^{n}f(z+c)-\Delta^{n}f(z)$, the results obtained are treating differences and divided differences $G(z):=\Delta f(z)/f(z)$, $G_{n}(z):=\Delta^{n}(z)/f(z)$ of $f$. The key results in this paper are as follows: (1) $G_{n}(z)$ has infinitely many zeros and infinitely many fixed points, provided $f$ is entire, $c=1$ and $\rho (f)<1/2$ and $\rho (f)\neq j/n$, $j=1,\ldots ,[n/2]$. (2) $G(z)$ has infinitely many zeros and infinitely many fixed points, whenever $f$ is entire with $\rho (f)<1$, and either $f$ has at most finitely many zeros whose difference is $=c$ or $\liminf_{j\rightarrow\infty}\vert z_{j+1}/z_{j}\vert =L>1$, where $\{z_{j}\}$ is the zero-sequence of $f$, arranged according to increasing moduli. (3) A similar result holds, if $f$ is entire of order $\rho (f)=1$ and with infinitely many zeros having the exponent of convergence $\lambda (f)<1$. (4) As for the case of $f$ meromorphic, a result similar to (2) follows by invoking corresponding conditions for the poles of $f$ as well. (5) Given a positive, non-decreasing function $\varphi :[1,\infty )\rightarrow [0,\infty)$ with $\lim_{r\rightarrow\infty}\varphi (r)=\infty$, there exists $f$ transcendental meromorphic such that $\limsup_{r\rightarrow\infty}(T(r,f)/r)<\infty$, $\limsup_{r\rightarrow\infty}(T(r,f)/\varphi (r)\log r)<\infty$ and that $\Delta f(z)$ has one fixed point only. The proofs rely on standard properties of meromorphic functions, Wiman-Valiron theory and some Nevanlinna theory.
[Ilpo Laine (Joensuu)]

MSC 2000:: *30D35 Distribution of values (one complex variable)

Keywords: complex difference; zero; fixed point

Citations: Zbl 1114.30028

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]