Skip to main content
SpringerLink
Log in

Some Results Regarding Partial Differential Polynomials and the Uniqueness of Meromorphic Functions in Several Variables

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this paper, we mainly investigate the value distribution of meromorphic functions in ℂm with its partial differential and uniqueness problem on meromorphic functions in ℂm and with its k-th total derivative sharing small functions. As an application of the value distribution result, we study the defect relation of a nonconstant solution to the partial differential equation. In particular, we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.

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

Access this article

Log in via an institution

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

References

  1. Brück R. On entire functions which share one value CM with their derivates. Result in Math, 1996, 30: 21–24

    Article  MATH  Google Scholar 

  2. Gundersen G G, Yang L Z. Entire functions that share one value with one or two of their derivates. J Math Anal Appl, 1998, 223: 245–260

    Article  Google Scholar 

  3. Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964

    MATH  Google Scholar 

  4. Hu P C, Li P, Yang C C. Unicity of Meromorphic Mappings. Berlin: Springer Science and Bussiness Media, 2013

    MATH  Google Scholar 

  5. Jin L. Theorems of Picard type for entire functions of several complex variables. Kodai Math J, 2003, 26: 221–229

    MATH  Google Scholar 

  6. Jin L. A unicity theorem for entire functions of several complex variables. Chin Ann Math, 2004, 25B: 483–492

    Article  MATH  Google Scholar 

  7. Li B Q. On Picard’s theorem. J Math Anal Appl, 2018, 460: 561–564

    Article  MATH  Google Scholar 

  8. Li B Q, Yang L. On Picard type theorems and entire solutions of differential equations. arXiv: 1809.05553v1

  9. Lü F. Theorems of Picard type for meromorphic function of several complex variables. Complex Variables and Elliptic Equations, 2013, 58: 1085–1092

    Article  MATH  Google Scholar 

  10. Ru M. Nevanlinna Theory and Its Relation to Diophantine Approxmation. Singapore: World Sci Publishing, 2001

    Book  MATH  Google Scholar 

  11. Vitter A. The lemma of the logarithmic derivative in several complex variables. Duke Math J, 1977, 44: 89–104

    Article  MATH  Google Scholar 

  12. Yi H X. A question of C.C. Yang on the uniqueness of entire functions. Kodai Math J, 1990, 13: 39–46

    Article  Google Scholar 

  13. Yi H X. Uniqueness theorems for meromorphic functions whose N-th derivatives share the same 1-points. Complex Variables and Elliptic Equations, 1997, 34: 421–436

    MATH  Google Scholar 

  14. Yang L Z. Further results on entire functions that share one value with their derivates. J Math Anal Appl, 1997, 212: 529–536

    Article  MATH  Google Scholar 

  15. Ye Z. A sharp form of Nevanlinna’s second main theorem of several complex variables. Math Z, 1996, 222: 81–95

    Article  Google Scholar 

  16. Yu K W. On entire and meromorphic functions that share small functions with thire derivatives. J Inequal Pure Appl Math, 2003, 4(1): Art 21

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, South China Agricultural University, Guangzhou, 510642, China

    Manli Liu  (刘曼莉) & Shaomei Fang  (房少梅)

  2. Department of Mathematics, Jinan University, Guangzhou, 510632, China

    Lingyun Gao  (高凌云)

Authors
  1. Manli Liu  (刘曼莉)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lingyun Gao  (高凌云)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shaomei Fang  (房少梅)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lingyun Gao  (高凌云).

Additional information

This work was partially supported by the NSFC (11271227, 11271161), the PCSIRT (IRT1264) and the Fundamental Research Funds of Shandong University (2017JC019).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, M., Gao, L. & Fang, S. Some Results Regarding Partial Differential Polynomials and the Uniqueness of Meromorphic Functions in Several Variables. Acta Math Sci 43, 821–838 (2023). https://doi.org/10.1007/s10473-023-0218-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-023-0218-0

Key words

2010 MR Subject Classification

Search

Navigation