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On zeros and fixed points of differences of meromorphic functions. (English) Zbl 1144.30012
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 [W. Bergweiler and J. K. Langley, Math. Proc. Camb. Philos. Soc. 142, No. 1, 133–147 (2007; Zbl 1114.30028)]. Denoting, for a transcendental meromorphic function $f$, ${\Delta }f\left(z\right):=f\left(z+c\right)-f\left(z\right)$, ${{\Delta }}^{n+1}f\left(z\right):={{\Delta }}^{n}f\left(z+c\right)-{{\Delta }}^{n}f\left(z\right)$, the results obtained are treating differences and divided differences $G\left(z\right):={\Delta }f\left(z\right)/f\left(z\right)$, ${G}_{n}\left(z\right):={{\Delta }}^{n}\left(z\right)/f\left(z\right)$ of $f$. The key results in this paper are as follows: (1) ${G}_{n}\left(z\right)$ has infinitely many zeros and infinitely many fixed points, provided $f$ is entire, $c=1$ and $\rho \left(f\right)<1/2$ and $\rho \left(f\right)\ne j/n$, $j=1,...,\left[n/2\right]$. (2) $G\left(z\right)$ has infinitely many zeros and infinitely many fixed points, whenever $f$ is entire with $\rho \left(f\right)<1$, and either $f$ has at most finitely many zeros whose difference is $=c$ or ${lim inf}_{j\to \infty }|{z}_{j+1}/{z}_{j}|=L>1$, where $\left\{{z}_{j}\right\}$ is the zero-sequence of $f$, arranged according to increasing moduli. (3) A similar result holds, if $f$ is entire of order $\rho \left(f\right)=1$ and with infinitely many zeros having the exponent of convergence $\lambda \left(f\right)<1$. (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 $\varphi :\left[1,\infty \right)\to \left[0,\infty \right)$ with ${lim}_{r\to \infty }\varphi \left(r\right)=\infty$, there exists $f$ transcendental meromorphic such that ${lim sup}_{r\to \infty }\left(T\left(r,f\right)/r\right)<\infty$, ${lim sup}_{r\to \infty }\left(T\left(r,f\right)/\varphi \left(r\right)logr\right)<\infty$ and that ${\Delta }f\left(z\right)$ has one fixed point only. The proofs rely on standard properties of meromorphic functions, Wiman-Valiron theory and some Nevanlinna theory.
##### MSC:
 30D35 Distribution of values (one complex variable); Nevanlinna theory
##### Keywords:
complex difference; zero; fixed point