Singular Direction and q-Difference Operator of Meromorphic Functions


We study the common singular direction problem of meromorphic function for q-difference version operator; some criterions of the existence of common singular direction have been established. Further, the common singular direction of solutions of q-difference equations is also discussed in this paper.

This is a preview of subscription content, access via your institution.


  1. 1.

    Anderson, J.M., Clunie, J.: Entire functions of finite order and lines of Julia. Math. Z. 112, 59–73 (1969)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Barnett, D.C., Halburd, R.G., Korhonen, R., Morgan, W.: Nevanlinna theory for the \(q\)-difference operator and meromorphic solutions of \(q\)-difference equations. Proc. R. Soc. Edinb. 137A, 457–474 (2007)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bergweiler, W., Langley, J.K.: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 142(1), 133–147 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bergweiler, W., Ishizaki, K., Yanagihara, N.: Growth of meromorphic solutions of some functional equations. Aequ. Math. 63(1–2), 140–151 (2002)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bergweiler, W., Rippon, P.J., Stallard, G.M.: Multiply connected wandering domains of entire functions. Proc. Lond. Math. Soc. 107(6), 1261–1301 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cao, T., Dai, H., Wang, J.: Nevanlinna theory for Jackson difference operators and entire solutions of \(q\)-difference equations. arXiv:1812.10014v2 [math.CV]. 4 Sept 2019

  7. 7.

    Chen, Z.X.: Complex Differences and Difference Equations. Science Press, Beijing (2014)

    Book  Google Scholar 

  8. 8.

    Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic of \(f(z+\eta )\) and difference equations in the complex plane. The Ramanujan J. 16, 105–129 (2008)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chiang, Y.M., Feng, S.J.: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Am. Math. Soc. 361(7), 3767–3791 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chiang, Y.M., Feng, S.J.: On the growth of logarithmic difference of meromorphic functions and a Wiman–Valiron estimate. Constr. Approx. 44, 313–326 (2016)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chuang, C.T.: Un théorème relatif aux directions de Borel des fonctions méromorphe d’ordre fini. C. R. Acad. Sci. 204, 951–952 (1937)

    MATH  Google Scholar 

  12. 12.

    Dai, C.J., Ji, S.Y.: Radial line of order \(\rho \) and its relation to the distribution of Borel directions. J. Shanghai Norm. Univ. 2, 16–24 (1980). (In Chinese)

    MATH  Google Scholar 

  13. 13.

    Drasin, D., Weitsman, A.: On the Julia directions and Borel directions of entire functions. Proc. Lond. Math. Soc. 32(2), 199–212 (1976)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gundersen, G., Heittokangas, J., Laine, I., Rieppo, J., Yang, D.G.: Meromorphic solutions of generalized Schröder equations. Aequ. Math. 63(1–2), 110–135 (2002)

    Article  Google Scholar 

  15. 15.

    Halburd, R.G., Korhonen, R.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477–487 (2006)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Halburd, R.G., Korhonen, R.: Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. (3) 94, 443–474 (2007)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Halburd, R.G., Korhonen, R., Tohge, K.: Holomorphic curves with shift-invariant hyperplane preimages. Trans. Am. Math. Soc. 366(8), 4267–4298 (2014)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hayman, W.K.: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964)

    Google Scholar 

  19. 19.

    Hiong, K.L.: Sur les fonctions entiéres et les fonctions méromorphes d’ordre infini. J. Math. Pures Appl. 14, 233–308 (1935)

    MATH  Google Scholar 

  20. 20.

    Ishizaki, K., Yanagihara, N.: Wiman–Valiron method for difference equations. Nogoya Math. J. 175, 75–102 (2004)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Ishizaki, K., Yanagihara, N.: Borel and Julia directions of meromorphic Schröder functions. Math. Proc. Camb. Philos. Soc. 139, 139–147 (2005)

    Article  Google Scholar 

  22. 22.

    Ishizaki, K., Yanagihara, N.: Borel and Julia directions of meromorphic Schröder functions II. Arch. Math. 87, 172–178 (2006)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Ishizaki, K., Yanagihara, N.: Singular directions of meromorphic functions of some non-autonomous Schröder equations, Potential theory in Matsue, 155-166, Adv. Stud. Pure Math. 44, Math. Soc. Japan, Tokyo (2006)

  24. 24.

    Milloux, H.: Sur les directions de Borel des fonctions entières, de leurs derivées et de leurs integrales. J’d Analyse Math. 1, 244–330 (1951)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Ostrowski, A.: Asymptotische Abschätzung des absoluten Betrages einer Funktion, die die Werte O und 1 nicht annimmt (German). Comment. Math. Helv. 5(1), 55–87 (1933)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Rauch, A.: Cas où une direction de Borel d’une fonction entière \(f(z)\) d’ordre fini est aussi direction de Borel pour \(f^{\prime }(z)\). C. R. Acad. Sci. 199, 1014–1016 (1934)

    MATH  Google Scholar 

  27. 27.

    Sun, D.C.: Common Borel directions of meromorphic functions of infinite order and its derivatives. Acta Math. Sin. 30(5), 641–647 (1987). (In Chinese)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Valiron, G.: Recherches sur le théorème de M. Borel dans la théorie des fonctions méromorphes. Acta Math. 52, 67–92 (1928)

    Article  Google Scholar 

  29. 29.

    Valiron, G.: Sur les directions de Borel des fonctions entières. Annali di Mat. 9, 273–285 (1931)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Wen, Z.T., Ye, Z.: Wiman–Valiron theorem for \(q\)-differences. Ann. Acad. Sci. Fenn. Math. 41, 305–312 (2016)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Yang, L.: Value Distribution Theory. Springer, Berlin (1993)

    MATH  Google Scholar 

  32. 32.

    Yang, L.: Common Borel directions of meromorphic functions and its derivatives. Sci. Sinica Special Issue (II), 91–104 (1979)

  33. 33.

    Yang, L., Zhang, Q.D.: New singular direction of meromorphic functions. Sci. Sin. Ser. A 27, 352–366 (1984)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Zhang, G.H.: Common Borel direcitons of meromorphic functions and its derivative or its integral. Acta Math. Sin. 20(2), 73–98 (1977). (In Chinese)

    Google Scholar 

  35. 35.

    Zhang, X.L.: A fundamental inequality for meromorphic functions in an angular doamin and its application. Acta Math. Sin. 10(3), 308–314 (1994)

    MathSciNet  Article  Google Scholar 

Download references


This research is supported by the National Natural Science Foundation of China (Grant No. 11861023, 11771090), and the Foundation of Science and Technology project of Guizhou Province of China (Grant No. [2018]5769-05).

Author information



Corresponding author

Correspondence to Jianren Long.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by V. Ravichandran.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Long, J., Qiao, J. & Yao, X. Singular Direction and q-Difference Operator of Meromorphic Functions. Bull. Malays. Math. Sci. Soc. 43, 3693–3709 (2020).

Download citation


  • Borel direction
  • Julia direction
  • Nevanlinna theory
  • q-difference operator
  • q-difference equation

Mathematics Subject Classification

  • Primary 30D35
  • Secondary 30D30
  • 39A13