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On the uniqueness theory of entire functions and their difference operators. (English) Zbl 1354.30020
The authors consider uniqueness problems for entire functions that share a small periodic entire functions with their shifts and difference operators. A meromorphic function \(a (z)\) is a small function of \(f (z)\) if \(T (r, a) = S (r, f )\), where \(S (r, f ) = o(T (r, f ))\), as \(r\to\infty\) outside of a possible exceptional set of finite logarithmic measure. Denote by \(S ( f )\) the family of all small functions with respect to \(f (z)\). Let \(f_c (z) = f (z + c)\) be a shift of \(f\) and \(\Delta_cf (z) = f (z + c)-f (z)\) be its difference operators. Let \(f (z)\) and \(g (z)\) be two meromorphic functions, and let \(a (z)\) be a small function with respect to \(f (z)\) and \(g (z)\). Then \(f (z)\) and \(g (z)\) share \(a (z)\) counting multiplicities (CM), provided that \(f (z)-a (z)\) and \(g (z)-a (z)\) have the same zeros with the same multiplicities. Let \(f (z)\) be a non-periodic entire function of finite order, and let \(a (z)\) (\(\not\equiv0\)) \(\in S ( f )\) be a periodic entire function with period \(c\). In particular, it is proved that, if \(f (z)\), \(\Delta_c f (z)\) and \(\Delta^2_cf (z)\) share \(a (z)\) CM, then \(\Delta_c f (z)\equiv f (z)\). The authors improve also some results due to B. Chen et al. [Abstr. Appl. Anal. 2012, Article ID 906893, 8 p. (2012; Zbl 1258.30010)]

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D15 Special classes of entire functions of one complex variable and growth estimates
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