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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

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