Singular Direction and q-Difference Operator of Meromorphic Functions

  • Jianren LongEmail author
  • Jianyong Qiao
  • Xiao Yao


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.


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

Mathematics Subject Classification

Primary 30D35 Secondary 30D30 39A13 



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).


  1. 1.
    Anderson, J.M., Clunie, J.: Entire functions of finite order and lines of Julia. Math. Z. 112, 59–73 (1969)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bergweiler, W., Langley, J.K.: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 142(1), 133–147 (2007)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)CrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)zbMATHGoogle 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)zbMATHGoogle 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)MathSciNetCrossRefGoogle 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)CrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)zbMATHGoogle Scholar
  20. 20.
    Ishizaki, K., Yanagihara, N.: Wiman–Valiron method for difference equations. Nogoya Math. J. 175, 75–102 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ishizaki, K., Yanagihara, N.: Borel and Julia directions of meromorphic Schröder functions. Math. Proc. Camb. Philos. Soc. 139, 139–147 (2005)CrossRefGoogle Scholar
  22. 22.
    Ishizaki, K., Yanagihara, N.: Borel and Julia directions of meromorphic Schröder functions II. Arch. Math. 87, 172–178 (2006)MathSciNetCrossRefGoogle 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)Google Scholar
  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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)zbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Valiron, G.: Recherches sur le théorème de M. Borel dans la théorie des fonctions méromorphes. Acta Math. 52, 67–92 (1928)CrossRefGoogle Scholar
  29. 29.
    Valiron, G.: Sur les directions de Borel des fonctions entières. Annali di Mat. 9, 273–285 (1931)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wen, Z.T., Ye, Z.: Wiman–Valiron theorem for \(q\)-differences. Ann. Acad. Sci. Fenn. Math. 41, 305–312 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yang, L.: Value Distribution Theory. Springer, Berlin (1993)zbMATHGoogle Scholar
  32. 32.
    Yang, L.: Common Borel directions of meromorphic functions and its derivatives. Sci. Sinica Special Issue (II), 91–104 (1979)Google Scholar
  33. 33.
    Yang, L., Zhang, Q.D.: New singular direction of meromorphic functions. Sci. Sin. Ser. A 27, 352–366 (1984)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020

Authors and Affiliations

  1. 1.School of Mathematical SciencesGuizhou Normal UniversityGuiyangPeople’s Republic of China
  2. 2.School of Computer Sciences and School of SciencesBeijing University of Posts and TelecommunicationsBeijingPeople’s Republic of China
  3. 3.School of Mathematical Sciences and LPMCNankai UniversityTianjinPeople’s Republic of China

Personalised recommendations