# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964. · Zbl 0115.06203 [2] H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, China, 1995. [3] W. K. Hayman, Research Problems in Function Theory, The Athlone Press University of London, London, UK, 1967. · Zbl 0158.06301 [4] W. K. Hayman, “Picard values of meromorphic functions and their derivatives,” Annals of Mathematics (2), vol. 70, pp. 9-42, 1959. · Zbl 0088.28505 · doi:10.2307/1969890 [5] J. Clunie, “On a result of Hayman,” Journal of the London Mathematical Society, vol. 42, pp. 389-392, 1967. · Zbl 0169.40801 · doi:10.1112/jlms/s1-42.1.389 [6] C. C. Yang and X. H. Hua, “Uniqueness and value-sharing of meromorphic functions,” Annales Academiæ Scientiarum Fennicæ Mathematica, vol. 22, no. 2, pp. 395-406, 1997. · Zbl 0890.30019 · emis:journals/AASF/Vol22/vol22.html · eudml:228963 [7] X.-G. Qi, L. Z. Yang, and K. Liu, “Uniqueness and periodicity of meromorphic functions concerning the difference operator,” Computers & Mathematics with Applications, vol. 60, no. 6, pp. 1739-1746, 2010. · Zbl 1202.30045 · doi:10.1016/j.camwa.2010.07.004 [8] Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of f(z+\eta ) and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105-129, 2008. · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1 [9] J.-F. Xu, Q. Han, and J.-L. Zhang, “Uniqueness theorems of meromorphic functions of a certain form,” Bulletin of the Korean Mathematical Society, vol. 46, no. 6, pp. 1079-1089, 2009. · Zbl 1186.30037 · doi:10.4134/BKMS.2009.46.6.1079 [10] H. X. Yi, “Meromorphic functions that share one or two values,” Complex Variables and Elliptic Equations, vol. 28, no. 1, pp. 1-11, 1995. · Zbl 0841.30027 [11] R. G. Halburdand and R. J. Korhonen, “Nevanlinna theory for the difference operator,” Annales Academiæ Scientiarium Fennicæ, vol. 31, no. 2, pp. 463-478, 2006. · Zbl 1108.30022 · eudml:127044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.