## 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 $$a\in \mathbb C$$ 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 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 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Zbl 0357.30007
Full Text:

### References:

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