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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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##### References:
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