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Uniqueness theorems on difference monomials of entire functions. (English) Zbl 1247.30047
Summary: The aim of this paper is to discuss the uniqueness of the difference monomials $$f^n f(z + c)$$. It is assumed that $$f$$ and $$g$$ are transcendental entire functions with finite order and $$E_{k)}(1, f^n f(z + c)) = E_{k)}(1, g^n g(z + c))$$, where $$c$$ is a nonzero complex constant and $$n, k$$ are integers. It is proved that $$fg = t_1$$ or $$f = t_2g$$ for some constants $$t_2$$ and $$t_3$$ which satisfy $$t^{n+1}_2 = 1$$ and $$t^{n+1}_3 = 1$$, if $$k = 1$$ and one of the following holds: (i) $$n \geq 6$$ and $$k = 3$$, (ii) $$n \geq 7$$ and $$k = 2$$, and (iii) $$n \geq 10$$. It is an improvement of the result of X.-G. Qi, L.-Z. Yang and K. Liu [Comput. Math. Appl. 60, No. 6, 1739–1746 (2010; Zbl 1202.30045)].

##### MSC:
 30D20 Entire functions of one complex variable, general theory
##### Keywords:
difference monomials; uniqueness; entire functions
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##### References:
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