Skip to main content
SpringerLink
Log in

Complex Oscillation of Linear Differential Equations with Analytic Coefficients of [p, q]-Order in the Unit Disc

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

In this paper, the authors study the interaction between the analytic coefficients of \({[p,q]}\)-order and the solutions of higher order linear differential equations in the unit disc, and add to the complex oscillation theory in the unit disc.

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. Bank, S.: A general theorem concerning the growth of solutions of first-order algebraic differential equations. Compos. Math. 25, 61–70 (1972)

    MATH  MathSciNet  Google Scholar 

  2. Belaïdi, B.: Growth and oscillation theroy of \([p, q]\)-order analytic solutions of linear equations in the unit disc. J. Math. Anal. 3(1), 1–11 (2012)

    MathSciNet  Google Scholar 

  3. Belaïdi, B.: Growth of solutions to linear equations with analytic coefficients of \([p, q]\)-order in the unit disc. Electron. J. Diff. Equ. 156, 1–11 (2011)

    MathSciNet  Google Scholar 

  4. Benbourenane, D., Sons, L.R.: On global solutions of complex diffenrential equations in the unit disk. Complex Var. Ell. Equ. 49, 913–925 (2004)

    MATH  MathSciNet  Google Scholar 

  5. Cao, T.B., Yi, H.X.: The growth of solutions of complex differential equations in the unit disc. J. Math. Anal. Appl. 319, 278–294 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, Z.X., Shon, K.H.: The growth of solutions of differential equations with coefficients of small growth in the unit disc. J. Math. Anal. Appl. 297, 285–304 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chyzhykov, I., Gundersen, G., Heittokangas, J.: Linear differential equations and logarithmic derivate of estimates. Proc. Lond. Math. Soc. 86, 735–754 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  9. Hayman, W.: On the characteristic of functions meromorphic in the unit disk and of their integrals. Acta Math. 112, 181–214 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  10. Heittokangas, J.: On complex differential equations in the unit disc. Ann. Acad. Sci. Fenn. Math. Diss. 122, 1–54 (2000)

    MathSciNet  Google Scholar 

  11. Heittokangas, J., Korhonen, R., Rättyä, J.: Fast growing solutions of linear differential equations in the unit disc. Results Math. 49, 265–278 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Heittokangas, J., Korhonen, R., Rättyä, J.: Generalized logarithmic derivative estimates of Gol’dberg–Grinshtein type. Bull. Lond. Math. Soc. 36, 105–114 (2004)

    Article  MATH  Google Scholar 

  13. Heittokangas, J., Korhonen, R., Rättyä, J.: Growth estimates for solutions of linear differential equations. Ann. Acad. Sci. Fenn. Math. 1 29, 233–246 (2004)

  14. Heittokangas, J., Korhonen, R., Rättyä, J.: Linear differential equations with coefficients in weighted bergman and hardy space. Trans. Am. Math. Soc. 360, 1035–1055 (2008)

    Article  MATH  Google Scholar 

  15. Heittokangas, J., Korhonen, R., Rättyä, J.: Linear differential equations with solutions in dirichlet type subspace of the hardy space. Nagoya Math. J. 187, 91–113 (2007)

    MATH  MathSciNet  Google Scholar 

  16. Heittokangas, J., Wen, Z.T.: Functions of finite logarithmic order in the unit disc, part I. J. Math. Anal. Appl. 415(1), 435–461 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Korhonen, R., Rättyä, J.: Finite order solutions of linear differential equations in the unit disc. J. Math. Anal. Appl. 349, 43–54 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Korhonen, R., Rättyä, J.: Linear differential equations in the unit disc with analytic solutions of finite order. Proc. Am. Math. Soc. 135, 1355–1363 (2007)

    Article  MATH  Google Scholar 

  19. Laine, I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin (1993)

    Book  Google Scholar 

  20. Latreuch, Z., Belaïdi, B.: Linear differential equations with analytic coefficients of \([p, q]\)-order in the unit disc. Sarajevo J. Math. 9(21), 71–84 (2013)

  21. Li, L.M., Cao, T.B.: Solutions for linear differential equations with meromorphic coefficients of \((p, q)\)-order in the plane. Electron. J. Diff. Equ. 2012(195), 1–15 (2012)

    Article  MATH  Google Scholar 

  22. Liu, J., Tu, J., Shi, L.Z.: Linear differential equations with coefficients of \((p, q)\)-order in the complex plane. J. Math. Anal. Appl. 372, 55–67 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pommerenke, Chr.: On the mean growth of the solutions of complex linear differential equations in the disc. Complex Var. Ell. Equ. 1(1), 23–38 (1982)

  24. Shea, D., Sons, L.R.: Value distribution theory for meromorphic functions of slow growth in the disk. Houston J. Math. 12(2), 249–266 (1986)

    MATH  MathSciNet  Google Scholar 

  25. Tu, J., Deng, G.T.: Growth of solutions of higher order linear differential equations with coefficient \(A_0\) being dominant. Acta Math. Sci. Ser. A. 30(4), 945–952 (2010) (in Chinese)

  26. Tsuji, M.: Potential Theory in Modern Function Theory. Reprint of the 1959 edition. Chelsea Publishing Co., New York (1975)

  27. Wittich, H.: Zur Theorie linearer Differentialgleichungen im Komplexen. Ann. Acad. Sci. Fenn. Ser. A. 379(1), 1–18 (1966)

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China

    Jin Tu

  2. Jiangshan Qinghu High School, Jiangshan, 324104, China

    Hai-Xia Huang

Authors
  1. Jin Tu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hai-Xia Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Tu.

Additional information

Communicated by Ilpo Laine.

This project is supported by the National Natural Science Foundation of China (11301233, 11261024, 11271045), the Natural Science Foundation of Jiangxi Province in China (20132BAB211002, 20122BAB211005) and the Foundation of Education Bureau of Jiangxi Province in China (GJJ14271, GJJ14272).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tu, J., Huang, HX. Complex Oscillation of Linear Differential Equations with Analytic Coefficients of [p, q]-Order in the Unit Disc. Comput. Methods Funct. Theory 15, 225–246 (2015). https://doi.org/10.1007/s40315-014-0103-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-014-0103-x

Keywords

Mathematics Subject Classification

Search

Navigation