Skip to main content
SpringerLink
Log in

Meromorphic Solutions for a Class of Differential Equations and Their Applications

  • Differential Equations
  • Published:
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) Aims and scope Submit manuscript

Abstract

In this note, we study the admissible meromorphic solutions for algebraic differential equation fnf' + Pn−1(f) = R(z)eα(z), where Pn−1(f) is a differential polynomial in f of degree ≤ n − 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless Pn−1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.

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. W. Bergweiler, “On the product of a meromorphic function and its derivative”, Bull. Hong Kong Math. Soc., 1, 97–101, 1997.

    MATH  Google Scholar 

  2. W. Bergweiler, A. Eremenko, “On the singularities of the inverse to a meromorphic function of finite order”, Rev. Mat. Iberoamericana, 11, 355–373, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. H. Chen, M. L. Fang, “The value distribution of f n f'”, Sci.China Ser.A, 38, 789–798, 1995.

    MathSciNet  Google Scholar 

  4. J. Clunie, “On integral and meromorphic functions”, J. Lond. Math. Soc., 37, 17–27, 1962.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Frank, W. Hennekemper, G. Polloczek, “Über die Nullstellen meromorpher Funktionen and ihrer Ableitungen”, Math. Ann., 225, 145–154, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Gackstatter, I. Laine, “Zur Theorie der gewöhnlichenDifferentialgleichungen im Komplexen”, Ann. Polon. Math., 38 (3), 259–287, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Grahl, “Hayman’s alternative and normal families of non-vanishing meromorphic functions”, Computational Methods and Function Theory, 2, 481–508, 2002.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. W. K. Hayman, “Picard values of meromorphic functions and their derivatives”, Ann. of Math., 70 (2), 9–42, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  10. W. K. Hayman, Research Problems in Function Theory, (Athlone Press University of London, London, 1967).

    MATH  Google Scholar 

  11. W. K. Hayman, J. Miles, “On the growth of a meromorphic function and its derivatives”, Complex Variables, 12, 245–260, 1989.

    MathSciNet  MATH  Google Scholar 

  12. Y. Z. He, I. Laine, “The Hayman-Miles theorem and the differential equation (y')n = R(z, y)”, Analysis, 10, 387–396, 1990.

    MathSciNet  MATH  Google Scholar 

  13. W. Hennekemper, “Über dieWertverteilung von (f k+1)(k)”, Math. Z., 177, 375–380, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  14. K. Ishizaki, N. Yanagihara, “Admissible solutions of algebraic differential equations”, Funkcial. Ekvac., 38, 433–442, 1995.

    MathSciNet  MATH  Google Scholar 

  15. I. Laine, Nevanlinna Theory and Complex Differential Equations (W. de Gruyter, Berlin, 1993).

    Book  MATH  Google Scholar 

  16. J. K. Langley, “On the zeros of ff''-b”, ResultsMath., 44, 130–140, 2003.

    MathSciNet  MATH  Google Scholar 

  17. L. W. Liao, Z. Ye, “On solutions to nonhomogengeous algebraic differential equations and their application”, J. Aust.Math. Soc., 97, 391–403, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  18. N. N. Liu, W. R. Lü, C. C. Yang, “On the meromorphic solutions of certain class of nonlinear differential equations”, Journal of Inequalities and Applications, 2015, 149, 2015.

    MathSciNet  Google Scholar 

  19. W. R. Lü, N. N. Liu, C. C. Yang, Caiping Zhuo, “Notes on the value distribution of ff (k)-b”, Kodai Math. J., 39, 500–509, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Mues, “Über ein Problem von Hayman”, Math. Z., 164, 239–259, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Ozawa, “Zeros of certain differential polynomials”, Analytic function theory of one complex variable, Pitman Res. NotesMath. Ser., 212, 199–225, 1989.

    MathSciNet  MATH  Google Scholar 

  22. C. C. Yang, L. Yang, Y. F. Wang, “On the zeros of (f (k))n f-a”, ChineseSci.Bull., 38, 2125–2128, 1993.

    Google Scholar 

  23. C. C. Yang, Z. Ye, “Estimates of the proximity function of differential polynomials”, Proc. Japan Acad. Ser. A Math. Sci., 83, 50–55, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  24. C. C. Yang, H. X. Yi, Uniqueness Theory of Meromorphic Functions (Science Press, Beijing, 2003).

    Book  MATH  Google Scholar 

  25. J. Zhang, L. Liao, “Admissible meromorphic solutions of algebraic differential equations”, J. Math. Anal. Appl., 397 (1), 225–232, 2013.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. China University of Petroleum, Qindao, Shandong, China

    W. R. Lü, F. Lü, L. Wu & J. Yang

Authors
  1. W. R. Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. R. Lü.

Additional information

Original Russian Text © W. R. Lü, F. Lü, L. Wu, J. Yang, 2018, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2018, No. 5, pp. 52–60.

Dedicated to Professor Chungchun Yang on the occasion of his 76th birthday.

This works was supported by NNSF of China Project ( No. 11601521) and the Fundamental Research Fund for Central Universities in China Project (Nos. 15CX05061A & 18CX02048A).

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lü, W.R., Lü, F., Wu, L. et al. Meromorphic Solutions for a Class of Differential Equations and Their Applications. J. Contemp. Mathemat. Anal. 53, 260–265 (2018). https://doi.org/10.3103/S1068362318050023

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1068362318050023

Keywords

MSC2010 numbers

Search

Navigation