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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 $\lambda \left(1/f\right)<\rho \left(f\right)$ and a non-zero complex constant $c$, if $n⩾2$ then $f{\left(z\right)}^{n}f\left(z+c\right)$ assumes every non-zero value $a\in ℂ$ 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 $c$, ${E}_{f\left(z\right)}\left({S}_{j}\right)={E}_{f\left(z+c\right)}\left({S}_{j}\right)$ $\left(j=1,2\right)$ implies $f\left(z\right)\equiv f\left(z+c\right)$. This gives an answer to a question of 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.

##### MSC:
 30D35 Distribution of values (one complex variable); Nevanlinna theory