Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Class of Meromorphic Functions Partially Shared Values with Their Differences or Shifts

  • Received : 2020.10.17
  • Accepted : 2021.08.10
  • Published : 2021.12.31
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Abstract

For a value s ∈ ℂ∪ {∞}, two meromorphic functions f and g are said to share the value s, CM, (or IM), provided that f(z)-s and g(z)-s have the same set of zeros, counting multiplicities, (respectively, ignoring multiplicities). We say that a meromorphic function f shares s ∈ Ŝ partially with a meromorphic function g if E(s, f) ⊆ E(s, g). It is easy to see that "partially shared values CM" are more general than "shared values CM". With the help of partially shared values, in this paper, we prove some uniqueness results between a non-constant meromorphic function and its generalized differences or shifts. We exhibit some examples to show that the result of Charak et al. [8] is not true for k = 2 or k = 3. We find some gaps in proof of the result of Lin et al. [24]. We not only correct these resuts, but also generalize them in a more convenient way. We give a number of examples to validate certain claims of the main results of this paper and also to show that some of conditions are sharp. Finally, we pose some open questions for further investigation.

Keywords

Acknowledgement

The author would like to thank the referees for their constructive comments that led to better presentation of the paper.

References

  1. M. B. Ahamed, On the Fermat-type difference equation f3(z)+[c1f(z+c)+c0f(z)]3 = eαz+β, J. Contemp. Math. Anal., 56(5)(2021), 255-269. https://doi.org/10.3103/S1068362321050022
  2. M. B. Ahamed, An investigation on the conjecture of Chen and Yi, Results Math., 74(3)(2019), 122. https://doi.org/10.1007/s00025-019-1045-4
  3. M. B. Ahamed, On the periodicity of meromorphic functions when sharing two sets IM, Stud. Univ. Bab'es-Bolyai Math., 64(3)(2019), 497-510. https://doi.org/10.24193/subbmath.2019.4.05
  4. M. B. Ahamed and G. Haldar, Uniqueness of difference-differential polynomials of meromorphic functions sharing a small function IM, J. Anal., (2021). https://doi.org/10.1007/s41478-021-00336-3.
  5. A. Banerjee and M. B. Ahamed, Results on meromorphic functions sharing two sets with its linear c-shift operators, J. Contemp. Math. Anal., 55(3)(2020), 143-155. https://doi.org/10.3103/S1068362320030024
  6. A. Banerjee and M. B. Ahamed, Uniqueness of meromorphic function with its shift operator under the purview of two or three shared sets, Math. Slovaca, 69(3)(2019), 557-572. https://doi.org/10.1515/ms-2017-0247
  7. A. Banerjee and M. B. Ahamed, On Some Sufficient Conditions for Periodicity of Meromorphic Function Under New Shared Sets, Filomat 33(18)(2019), 6055-6072. https://doi.org/10.2298/fil1918055b
  8. K. S. Charak, R. J. Korhonen and G. Kumar, A note on partial sharing of values of meromorphic function with their shifts, J. Math. Anal. Appl., 435(2)(2016), 1241-1248. https://doi.org/10.1016/j.jmaa.2015.10.069
  9. B. Q. Chen and Z. X. Chen, Meromorphic functions sharing two sets with its Difference Operator, Bull. Malays. Math. Sci. Soc., 35(3)(2012), 765-774.
  10. S. J. Chen, On Uniqueness of Meromorphic Functions and Their Difference Operators with Partially Shared Values, Comput. Methods Funct. Theory 18(3)(2018), 529-536. https://doi.org/10.1007/s40315-018-0238-2
  11. S. J. Chen and W. C. Lin, Periodicity and uniqueness of meromorphic functions concerning three sharing values, Houston J. Math. 2016, Preprint.
  12. S. J. Chen and A. Z. Xu, Periodicity and unicity of meromorphic functions with three shared values, J. Math. Anal. Appl., 385(1)(2012), 485-490. https://doi.org/10.1016/j.jmaa.2011.06.072
  13. 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(1)(2008), 105-129. https://doi.org/10.1007/s11139-007-9101-1
  14. Y.-M. Chiang and S. Feng, Nevanlinna theory of the AskeyWilson divided difference operator, Advances Math., 329(2018), 217-272. https://doi.org/10.1016/j.aim.2018.02.006
  15. A. Goldberg and I. Ostrovskii, Value Distribution of Meromorphic Functions, Transl. Math. Monogr., vol. 236, American Mathematical Society, Providence, RI, 2008, translated from the 1970 Russian original by Mikhail Ostrovskii, with an appendix by Alexandre Eremenko and James K. Langley.
  16. W. K. Hayman, Meromorphic functions, The Clarendon Press, Oxford (1964).
  17. 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(2)(2006), 477-487. https://doi.org/10.1016/j.jmaa.2005.04.010
  18. R. G. Halburd and R. j. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31(2)(2006), 463-478.
  19. R. G. Halburd, R. Korhonen and K. Tohge, Holomorphic curves with shift invariant hyperplane preimages, Trans. Amer. Math. Soc., 366(8)(2014), 4267-4298. https://doi.org/10.1090/S0002-9947-2014-05949-7
  20. J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ., 56(1-4)(2001), 81-92. https://doi.org/10.1080/17476930903394770
  21. J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl., 355(1)(2009), 352-363. https://doi.org/10.1016/j.jmaa.2009.01.053
  22. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
  23. X. M. Li and H. X. Yi, Meromorphic functions sharing four values with their difference operators or shifts, Bull. Korean Math. Soc., 53(4)(2016), 1213-1235. https://doi.org/10.4134/BKMS.b150609
  24. W. C. Lin, X. Q. Lin and A. Wu, Meromorphic functions partially shared values with their shifts, Bull. Korean Math. Soc., 55(2)(2018), 469-478. https://doi.org/10.4134/BKMS.b170072
  25. K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl., 359(1)(2009), 384-393. https://doi.org/10.1016/j.jmaa.2009.05.061
  26. K. Liu and L. Z. Yang, Value distribution of the difference operator, Arch. Math. (Basel), 92(3)(2009), 270-278. https://doi.org/10.1007/s00013-009-2895-x
  27. K. Liu, T. B. Cao and H. Z. Cao, Entire solutions of Fermat type differential-difference equations, Arch. Math., 99(2012), 147-155. https://doi.org/10.1007/s00013-012-0408-9
  28. F. Lu, Q. Han and W. Lu, On the unicity of meromorphic solutions to difference equations of Malmquist type, Bull. Aust. Math. Soc., 93(2016), 92-98. https://doi.org/10.1017/S0004972715000787
  29. A. Z. Mohonko, The nevanlinna characteristics of certain meromorphic functions, Teor. Funktsii Funktsional. Anal. i Prilozhen, 14(1971), 83-87 (Russian).
  30. R. Nevanlinna, Le thorme de PicardBorel et la thorie des fonctions mromorphes, GauthiersVillars, Paris, 1929.
  31. G. Valiron, Sur la derivee des fonctions algebroides., Bull. Soc. Math. France, 59(1931), 17-39. https://doi.org/10.24033/bsmf.1170
  32. L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterranean J. Math., 15(2018), Article number: 227.
  33. K. Yamanoi, The second main theorem for small functions and related problems, Acta Math., 192(2)(2004), 225-294. https://doi.org/10.1007/BF02392741
  34. N. Yanagihara, Meromorphic solutions of some difference equations, Funkcialaj Ekvacioj, 23(1980), 309-326.
  35. C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003.
  36. J. Zhang, Value distribution and sets of difference of meromorphic functions, J. Math. Anal. Math., 367(2)(2010), 401-408. https://doi.org/10.1016/j.jmaa.2010.01.038