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Value distribution and shared sets of differences of meromorphic functions. (English) Zbl 1188.30044
Summary: We investigate value distribution and uniqueness problems for difference polynomials of meromorphic functions. In particular, we show that, for a finite order transcendental meromorphic function $f$ with $\lambda (1/f)<\rho (f)$ and a non-zero complex constant $c$, if $n\geqslant 2$ then $f(z)^nf(z+c)$ assumes every non-zero value $a\in \Bbb C$ infinitely often. This research also shows that there exist two sets $S_{1}$ with $9$ (resp. $5$) elements and $S_{2}$ with $1$ element such that, for a finite order non-constant meromorphic (resp. entire) function $f$ and a non-zero complex constant $c$, $E_{f(z)}(S_j)=E_{f(z+c)}(S_j)$ $ (j=1,2)$ implies $f(z)\equiv f(z+c)$. This gives an answer to a question of {\it F. Gross} [Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 51--67 (1977; Zbl 0357.30007)] concerning a finite order meromorphic function $f$ and its shift.

30D35Distribution of values (one complex variable); Nevanlinna theory
Full Text: DOI
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