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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 \(a\in \mathbb C\) 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

Citations:

Zbl 0357.30007
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References:

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