Skip to main content
SpringerLink
Log in

An Investigation on the Conjecture of Chen and Yi

  • Published:
Results in Mathematics Aims and scope Submit manuscript

A Correction to this article was published on 16 July 2019

This article has been updated

Abstract

In the paper, we have investigated on a conjecture posed by Chen and Yi (Results Math 63:557–565, 2013) concerning the uniqueness problem of meromorphic functions f sharing three distinct values with their difference \({\mathcal {L}}_c(f) \). We have proved the conjecture for finite ordered meromorphic functions. Some examples have been exhibited in the paper to show that the main result is true also for the meromorphic function of infinite order, but we are unable to prove our results for the function of infinite order, and hence we conjecture it. The main results in the paper also generalized a result of Zhang and Liao (Sci China Math 57(10):2143–2152, 2014). This research also shows that when a meromorphi function f satisfies a certain relation of the type \( {\mathcal {L}}_c(f)\equiv f \), then it can be found the class of all such functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

  • 16 July 2019

    In the original publication, Examples 1.9 and 1.12 are exhibited inappropriately for infinite-order entire functions.

References

  1. Bergweiler, W., Langley, J.K.: Zeros of difference of meromorphic functions. Math. Proc. Camb. Philos. Soc. 142, 133–147 (2007)

    Article  MathSciNet  Google Scholar 

  2. Brück, R.: On entire functions which share one value CM with their derivative. Results Math. 30, 21–24 (1996)

    Article  MathSciNet  Google Scholar 

  3. Chen, Z.X.: Some results on difference Riccati equations. Acta Math. Sin. Eng. Ser. 27, 1091–1100 (2011)

    Article  MathSciNet  Google Scholar 

  4. Chen, Z.X., Yi, H.X.: On sharing values of meromorphic functions and their differences. Results Math. 63, 557–565 (2013)

    Article  MathSciNet  Google Scholar 

  5. Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic of \( f(z+c) \) and difference equation in the complex plane. Ramanujan J. 16, 105–129 (2008)

    Article  MathSciNet  Google Scholar 

  6. Gundersen, G.: Meromorphic function share four values. Trans. Am. Math. Soc. 277, 545–567 (1983)

    Article  MathSciNet  Google Scholar 

  7. Gundersen, G.: Correction to meromorphic functions that share four values. Trans. Am. Math. Soc. 304, 847–850 (1987)

    MATH  Google Scholar 

  8. Halburd, R.G., Korhonen, R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31, 463–478 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Halburd, R.G., Korhonen, R.J.: Meromorphic solution of difference equations, integrability and the discrete. J. Phys. A 40, 1–38 (2007)

    Article  MathSciNet  Google Scholar 

  10. Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)

    MATH  Google Scholar 

  11. Heittokangas, J., Korhonen, R., Laine, I., et al.: Value sharing results for shifts of meromorphic functions, and sufficient condition for periodicity. J. Math. Anal. Appl. 355, 352–363 (2009)

    Article  MathSciNet  Google Scholar 

  12. Liu, K.: Zeros of difference polynomials of meromorphic functions. Results Math. 57, 365–376 (2010)

    Article  MathSciNet  Google Scholar 

  13. Liu, K., Yang, L.Z.: Value distribution of the difference operator. Arch. Math. 92, 270–278 (2009)

    Article  MathSciNet  Google Scholar 

  14. Lü, F., Lü, W.: Meromorphic functions sharing three values with their difference operators. Comput. Methods Funct. Theory. (2017). https://doi.org/10.1007/s40315-016-0188-5

  15. Mohonko, A.Z.: The Nevanlinna characteristics of certain meromorphic functions. Teor. Funktsii Funktsional. Anal. i Prilozhen 14, 83–87 (1971). (Russian)

    MathSciNet  Google Scholar 

  16. Mues, E.: Meromorphic functions sharing four values. Complex Var. Theory Appl. 12, 169–179 (1989)

    MathSciNet  MATH  Google Scholar 

  17. Nevanlinna, R.: Le Théoréme de Picard-Borel et la Théorie des Fonctions Méromorphes. Gauthier-Villars, Paris (1929)

    MATH  Google Scholar 

  18. Rubel, L., Yang, C.C.: Values shared by an entire function and its derivative. In: Lecture Notes in Mathematics, vol. 599, pp. 101–103. Springer, Berlin (1977)

  19. Valiron, G.: Sur la dérivée des fonctions algébroïdes. Bull. Soc. Math. France 59, 17–39 (1931)

    Article  MathSciNet  Google Scholar 

  20. Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishing Group, Dordrecht (2003)

    Book  Google Scholar 

  21. Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Science Press, Beijing (2006)

    Google Scholar 

  22. Yang, L.: Value Distribution Theory. Springer, Berlin (1993)

    MATH  Google Scholar 

  23. Zhang, J., Liao, L.W.: Entire function sharing some values with their difference operators. Sci. China Math. 57(10), 2143–2152 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kalipada Ghosh Tarai Mahavidyalaya, Bagdogra, West Bengal, 734014, India

    Molla Basir Ahamed

Authors
  1. Molla Basir Ahamed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Molla Basir Ahamed.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahamed, M.B. An Investigation on the Conjecture of Chen and Yi. Results Math 74, 122 (2019). https://doi.org/10.1007/s00025-019-1045-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-019-1045-4

Keywords

Mathematics Subject Classification

Search

Navigation