Advertisement

Chinese Annals of Mathematics, Series B

, Volume 41, Issue 1, pp 27–36 | Cite as

Properties of Complex Oscillation of Solutions of a Class of Higher Order Linear Differential Equations

  • Jianren Long
  • Yezhou LiEmail author
Article
  • 4 Downloads

Abstract

Let A(z) be an entire function with \(\mu (A) < \tfrac{1}{2}\) such that the equation f(k) + A(z)f = 0, where k ≥ 2, has a solution f with λ(f) < μ(A), and suppose that A1 = A + h, where h ≢ 0 is an entire function with ρ(h) < μ(A). Then g(k) + A1(z)g = 0 does not have a solution g with λ(g) < ∞.

Keywords

Complex differential equations Entire function Order of growth Exponent of convergence of the zeros 

2000 MR Subject Classification

34M10 30D35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Alotaibi, A., On complex oscillation theory, Results Math., 47, 2005, 165–175.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Bank, S. and Laine, I., On the oscillation theory of f″ + Af = 0 where A is entire, Trans. Amer. Math. Soc., 273(1), 1982, 351–363.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Bank, S., Laine, I. and Langley, J. K., On the frequency of zeros of solutions of second order linear differential equations, Results Math., 10, 1986, 8–24.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Bank, S. and Langley, J. K., Oscillation theory for higher order linear differential equations with entire coefficients, Complex Var. Theory Appl., 16(2–3), 1991, 163–175.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Barry, P., Some theorems related to the cos πρ theorem, Proc. London Math. Soc., 21(3), 1970, 334–360.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Chen, Z. X. and Gao, S. A., On complex oscillation property of solutions for higher-order periodic differential equations, J. Inequal. Appl., 2007, 2007, 13 pages, Article ID: 58189.Google Scholar
  7. [7]
    Chen, Z. X. and Yang, C. C., Some further results on the zeros and growths of entire solutions of second order linear differential equations, Kodai Math. J., 22, 1999, 273–285.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Chen, Z. X. and Yang, C. C., Quantitative estimation on the zeros and growths of entire solutions of linear differential equations, Complex Var. Theory Appl., 42(2), 2000, 119–133.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Chiang, Y. M., Oscillation results on y″ + Ay = 0 in the complex domain with transcendental entire coefficients which have extremal deficiencies, Proc. Edinburgh Math. Soc., (2), 38(1), 1995, 13–34.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Chiang, Y. M., Laine, I. and Wang, S. P., Oscillation results for some linear differential equations, Math. Scand, 77(2), 1995, 209–224.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Hayman, W., Meromorphic Functions, Clarendon Press, Oxford, 1964.zbMATHGoogle Scholar
  12. [12]
    Hille, E., Ordinary Differential Equations in the Complex Domain, Dover Publications Inc., Mineola, NY, 1997.zbMATHGoogle Scholar
  13. [13]
    Laine, I., Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter & Co., Berlin, New York, 1993.CrossRefGoogle Scholar
  14. [14]
    Laine, I. and Wu, P. C., On the osillation of certain second order linear differential equations, Rev. Roumaine Math. Pures Appl., 44(4), 1999, 609–615.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Langley, J. K., Some oscillation theorems for higher order linear differential equations with entire coefficients of small growth, Results Math., 20, 1991, 517–529.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Langley, J. K., On entire solutions of linear differential equations with one dominant coefficients, Analysis, 15(2), 1995, 187–204.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Langley, J. K., Postgraduate notes on complex analysis, http://www.maths.nott-ingham.ac.uk/personal/jkl/pg1.pdf.
  18. [18]
    Latreuch, Z. and Belaïdi, B., Growth and oscillation of differential polynomials generated by complex differential equations, Electro. J. Differential Equations, 16, 2013, 14 pages.Google Scholar
  19. [19]
    Li, Y. Z. and Wang, J., Oscillation of solutions of linear differential equations, Acta. Math. Sinica, 24(1), 2008, 167–176.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Rossi, J., Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc., 97, 1986, 61–66.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Yang, L., Value Distribution Theory, Springer-Verlag, Berlin; Science Press Beijing, Beijing, 1993.zbMATHGoogle Scholar

Copyright information

© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2020

Authors and Affiliations

  1. 1.School of Computer Science and School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.School of Mathematical ScienceGuizhou Normal UniversityGuiyangChina
  3. 3.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina

Personalised recommendations