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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 2 g for some constants t 2 and t 3 which satisfy t 2 n+1 =1 and t 3 n+1 =1, if k=1 and one of the following holds: (i) n6 and k=3, (ii) n7 and k=2, and (iii) n10. 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)].
30D20General theory of entire functions