Value distribution for nth difference operator of meromorphic functions with maximal deficiency sum

  • Subhas S. Bhoosnurmath
  • Renukadevi S. Dyavanal
  • Mahesh Barki
  • Ashok Rathod
Original Research Paper


In this paper, we obtain the relationship between the characteristic function of meromorphic function having maximal deficiency sum and its higher order exact difference. We improve and generalise several results of Zhaojun Wu (Journal of Inequalities and Applications 530, [3]) to a great extent. We obtain an analogue of Shah and Singh (Mathematische Zeitschrift 65:171–174, [4]) in an improved form for \(\Delta _{c}f\). Also we establish a lower bound for \(K(\Delta _{c}f)=\limsup \limits _{r\rightarrow \infty }\frac{N\left( r,\Delta _{c}f\right) +N\left( r,\frac{1}{\Delta _{c}f}\right) }{T(r,\Delta _{c}f)}\). Under certain conditions we also show that \(K(\Delta _{c}^{n}f)=0\), for any integer \(n\ge 1\).


Nevanlinna theory Complex difference equations Meromorphic functions Maximal deficiency sum Small functions 

Mathematics Subject Classification




The first and third authors are supported by the SERB(DST) Project No. SB/S4/MS/842/2013, dated:10/12/2015. We are grateful to the referees for valuable suggestions and comments.

Compliance with ethical standards

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Human/animals participants

The authors declare that there is no research involving human participants and/or animals in the contained of this paper.


  1. 1.
    Hayman, W.K. 1964. Meromorphic functions. Oxford: Clarendon Press.zbMATHGoogle Scholar
  2. 2.
    Yang, C.C., and H.X. Yi. 2004. Uniqueness theory of meromorphic functions. Dordrecht: Kluwer.Google Scholar
  3. 3.
    Zhaojun Wu. 2013. Value distribution for difference operator of meromorphic functions with maximal deficiency sum. Journal of Inequalities and Applications 530.Google Scholar
  4. 4.
    Shah, S.M., and S.K. Singh. 1956. On the derivative of a meromorphic function with maximum defect. Mathematische Zeitschrift 65: 171–174.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Singh, S.K., and H.S. Gopalakrishna. 1971. Exceptional values of entire and meromorphic functions. Mathematische Annalen 191: 121–142.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bergweiler, W., and J.K. Langley. 2007. Zeros of differences of meromorphic functions. Mathematical Proceedings of the Cambridge Philosophical Society 142: 133–147.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Halburd, R.G., and R.J. Korhonen. 2006. Nevanlinna theory for the difference operator. Annales-Academiae Scientiarum Fennicae Mathematica 31 (2): 463–478.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Qi, Xiao-Guang, Lian-Zhong Yang, and Kai Liu. 2010. Uniqueness and periodicity of meromorphic functions concerning the difference operator. Computer and Mathematics with Applications 60: 1739–1746.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Heittokangas, J., R. Korhonen, I. Laine, and J. Rieppo. 2009. Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity. Journal of Mathematical Analysis and Applications 355: 352–363.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Forum D'Analystes, Chennai 2018

Authors and Affiliations

  • Subhas S. Bhoosnurmath
    • 1
  • Renukadevi S. Dyavanal
    • 1
  • Mahesh Barki
    • 1
  • Ashok Rathod
    • 1
  1. 1.Department of MathematicsKarnatak UniversityDharwadIndia

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