Acknowledgement
The authors wish to thank the referee for his/her valuable suggestions towards the improvement of this paper. The second author is thankful to Council of Scientific and Industrial Research (India) for their financial support under File No: 09/106(0191)/2019-EMR-I.
References
- A. Banerjee and S. Bhattacharyya, Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets, Adv. Difference Equ. 2019 (2019), Paper No. 509, 23 pp. https://doi.org/10.1186/s13662-019-2425-5
- K. S. Charak, R. J. Korhonen, and G. Kumar, A note on partial sharing of values of meromorphic functions with their shifts, J. Math. Anal. Appl. 435 (2016), no. 2, 1241-1248. https://doi.org/10.1016/j.jmaa.2015.10.069
- C.-X. Chen and Z.-X. Chen, A note on entire functions and their differences, J. Inequal. Appl. 2013 (2013), 587, 17 pp. https://doi.org/10.1186/1029-242x-2013-587
- Z.-X. Chen and H.-X. Yi, On sharing values of meromorphic functions and their differences, Results Math. 63 (2013), no. 1-2, 557-565. https://doi.org/10.1007/s00025-011-0217-7
- Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129. https://doi.org/10.1007/s11139-007-9101-1
- R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477-487. https://doi.org/10.1016/j.jmaa.2005.04.010
- R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463-478.
- R. G. Halburd and R. J. Korhonen, Value distribution and linear operators, Proc. Edinb. Math. Soc. 57 (2014), 493-504. https://doi.org/10.1017/S0013091513000448
- R. Halburd, R. Korhonen, and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4267-4298. https://doi.org/10.1090/S0002-9947-2014-05949-7
- W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
- J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), no. 1, 352-363. https://doi.org/10.1016/j.jmaa.2009.01.053
- J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ. 56 (2011), no. 1-4, 81-92. https://doi.org/10.1080/17476930903394770
- Z.-B. Huang, Value distribution and uniqueness on q-differences of meromorphic functions, Bull. Korean Math. Soc. 50 (2013), no. 4, 1157-1171. https://doi.org/10.4134/BKMS.2013.50.4.1157
- Z.-B. Huang and R.-R. Zhang, Uniqueness of the differences of meromorphic functions, Anal. Math. 44 (2018), no. 4, 461-473. https://doi.org/10.1007/s10476-018-0306-x
- Y. Jiang and Z. Chen, Meromorphic functions share two values with its difference operator, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 63 (2017), no. 1, 169-175.
- X.-M. Li, H.-X. Yi, and C.-Y. Kang, Notes on entire functions sharing an entire function of a smaller order with their difference operators, Arch. Math. (Basel) 99 (2012), no. 3, 261-270. https://doi.org/10.1007/s00013-012-0425-8
- F. Lu and W. Lu, Meromorphic functions sharing three values with their difference operators, Comput. Methods Funct. Theory 17 (2017), no. 3, 395-403. https://doi.org/10.1007/s40315-016-0188-5
- X. Luo and W.-C. Lin, Value sharing results for shifts of meromorphic functions, J. Math. Anal. Appl. 377 (2011), no. 2, 441-449. https://doi.org/10.1016/j.jmaa.2010.10.055
- V. Noulorvang and D. T. Pham, On partial value sharing results of meromorphic functions with their shifts and its applications, Bull. Korean Math. Soc. 57 (2020), no. 5, 1083-1094. https://doi.org/10.4134/BKMS.b190483
- K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), no. 2, 225-294. https://doi.org/10.1007/BF02392741
- C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003.
- J. Zhang and L. Liao, Entire functions sharing some values with their difference operators, Sci. China Math. 57 (2014), no. 10, 2143-2152. https://doi.org/10.1007/s11425-014-4848-5
- L. Zhen, Meromorphic functions sharing three polynomials with their difference operators, J. Contemp. Math. Anal. 54 (2019), no. 5, 296-301; translated from Izv. Nats. Akad. Nauk Armenii Mat. 54 (2019), no. 5, 44-52. https://doi.org/10.3103/S1068362319050066