Advertisement

Applications of difference analogue of Cartan’s second main theorem for holomorphic curves

  • Renukadevi Sangappa DyavanalEmail author
Original Research Paper
  • 5 Downloads

Abstract

In this paper, we present refinement of earlier results on sufficient conditions for periodicity of a meromorphic function which are obtained by using the new version of Cartan’s second main theorem for the Casorati determinant.

Keywords

Nevanlinna theory Holomorphic curve Difference operator Casorati determinant Difference analogue of Cartan’s second main theorem 

Mathematics Subject Classification

30D35 32H30 

Notes

Compliance with ethical standards

Conflict of interest

The author is thankful to the referee for useful suggestions towards the improvement of the present paper. The author is supported by Ref. No. F.510/3/DRS-III/2016(SAP-I) Dated:29th Feb.2016, Department of Mathematics, Karnatak University, Dharwad.

Ethical approval

This article does not contain studies with human participants or animals by any of the authors.

References

  1. 1.
    Chiang, Y.M., and S.J. Feng. 2008. On the Nevanlinna characteristic of \(f(z+\eta )\) and difference equations in the complex plane. Ramanujan J 16: 105–129.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bhoosnurmath, S.S., R.S. Dyavanal, M. Barki, and A. Rathod. 2018. Value distribution for nth Difference operator of meromorphic functions with maximal deficiency sum. J Anal.  https://doi.org/10.1007/s41478-018-0130-5.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dyavanal, R.S., and M.M. Mathai. 2018. Value distribution of general diffential-difference polynomials of meromorphic functions. J Anal.  https://doi.org/10.1007/s41478-018-0155-9.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Halburd, R.G., and R. Korhonen. 2006. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314: 477–487.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Halburd, R.G., and R. Korhonen. 2006. Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31: 463–487.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Halburd, R.G., R.J. Korhonen, and K. Tohge. 2014. Holomorphic curves with shift-invariant hyperplane preimages. Trans. Amer. Math. Soc. 366: 4267–4298.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Heittokangas, J., R. Korhonen, I. Laine, and J. Rieppo. 2011. Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. 56: 81–92.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Heittokangas, J., R. Korhonen, I. Laine, J. Rieppo, and J. Zhang. 2009. Value sharing results for shifts of meromorphic function and sufficient conditions for periodicity. J. Math. Anal. Appl. 355: 352–363.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Laine, I. 1993. Nevanlinna theory and complex differential equations. Berlin, Newyork: Walter der Gruyter.CrossRefGoogle Scholar
  10. 10.
    Qi, X.G., L.Z. Yang, and K. Liu. 2010. Uniqueness and periodicity of meromorphic functions concerning the difference operator. Comp. Math. Appl. 60: 1739–1746.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Yang, C.C., and H.X. Yi. 2003. Uniqueness Theory of Meromorphic Functions. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
  12. 12.
    Yang, C.C., and H.X. Yi. 1995. Uniqueness Theory of Meromorphic Functions. Beijing: Chinese Original, Science Press.Google Scholar
  13. 13.
    Yang, L. 1993. Value Distribution Theory. Berlin: Springer-Verlang and Science Press.zbMATHGoogle Scholar

Copyright information

© Forum D'Analystes, Chennai 2019

Authors and Affiliations

  1. 1.Department of MathematicsKarnatak UniversityDharwadIndia

Personalised recommendations