Uniqueness theorems on entire functions and their difference operators or shifts. (English) Zbl 1258.30010

Uniqueness theory of meromorphic functions is an important part of the Nevanlinna theory. The authors study the uniqueness problems on entire functions and their difference operators or shifts. The main result is a difference analogue of a result of G. Jank et al. [Complex Variables, Theory Appl. 6, 51–71 (1986; Zbl 0603.30037)], which is concerned with the uniqueness of the entire function sharing one finite value with its derivatives. Moreover, two relative results are proved, and examples are provided for their results.


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D10 Representations of entire functions of one complex variable by series and integrals


Zbl 0603.30037
Full Text: DOI


[1] W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964. · Zbl 0141.07802 · doi:10.1007/BF02391770
[2] I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1993. · Zbl 0827.42018
[3] C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. · Zbl 1232.06020
[4] L. A. Rubel and C.-C. Yang, “Values shared by an entire function and its derivative,” in Complex Analysis, vol. 599 of Lecture Notes in Mathematics, pp. 101-103, Springer, Berlin, Germany, 1977. · Zbl 0362.30026
[5] G. Jank, E. Mues, and L. Volkmann, “Meromorphe Funktionen, die mit ihrer ersten und zweiten Ableitung einen endlichen Wert teilen,” Complex Variables, Theory and Application, vol. 6, no. 1, pp. 51-71, 1986. · Zbl 0603.30037
[6] W. Bergweiler and J. K. Langley, “Zeros of differences of meromorphic functions,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 142, no. 1, pp. 133-147, 2007. · Zbl 1114.30028 · doi:10.1017/S0305004106009777
[7] Z.-X. Chen, “Value distribution of products of meromorphic functions and their differences,” Taiwanese Journal of Mathematics, vol. 15, no. 4, pp. 1411-1421, 2011. · Zbl 1239.30015
[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] R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 477-487, 2006. · Zbl 1085.30026 · doi:10.1016/j.jmaa.2005.04.010
[10] R. G. Halburd and R. J. Korhonen, “Nevanlinna theory for the difference operator,” Annales Academiæ Scientiarium Fennicæ. Mathematica, vol. 31, no. 2, pp. 463-478, 2006. · Zbl 1108.30022
[11] J.-L. Zhang, “Value distribution and shared sets of differences of meromorphic functions,” Journal of Mathematical Analysis and Applications, vol. 367, no. 2, pp. 401-408, 2010. · Zbl 1188.30044 · doi:10.1016/j.jmaa.2010.01.038
[12] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang, “Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 352-363, 2009. · Zbl 1180.30039 · doi:10.1016/j.jmaa.2009.01.053
[13] S. Li and Z.-S. Gao, “Entire functions sharing one or two finite values CM with their shifts or difference operators,” Archiv der Mathematik, vol. 97, no. 5, pp. 475-483, 2011. · Zbl 1243.30068
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.