Bulletin of the Iranian Mathematical Society

, Volume 44, Issue 1, pp 19–41 | Cite as

Meromorphic Functions Having the Same Inverse Images of Four Values on Annuli

  • Si Duc QuangEmail author
  • Tran An Hai
  • Nguyen Thi Thanh Hien
  • Ha Huong Giang
Original Paper


In this paper, we extend and improve the four-value theorems of Nevanlinna and Fujimoto to the case of meromorphic functions on the annuli. For detail, we will prove that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicities truncated by two and other three values regardless of multiplicities. We also show that if four admissible meromorphic functions on an annulus share four values regardless of multiplicities then two of them must coincide. Moreover, in our result, the inverse images of these values by the functions with multiplicities more than a certain number do not need to be counted.


Meromorphic function Nevanlinna theory Annulus 

Mathematics Subject Classification

Primary 30D35 Secondary 32H30 



This paper is completed while the first author was working at Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the institute for the support. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 101.04-2018.01.


  1. 1.
    Axler, S.: Harmonic functions from a complex analysis viewpoint. Am. Math. Monthly 93, 246–258 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Banerjee, A.: Weighted sharing of a small function by a meromorphic function and its derivative. Comput. Math. Appl. 53, 1750–1761 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bhoosnurmath, S.S., Dyavanal, R.S.: Uniqueness and value-sharing of meromorphic functions. Comput. Math. Appl. 53, 1191–1205 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cao, T.B., Yi, H.X., Xu, H.Y.: On the multiple values and uniqueness of meromorphic functions on annuli. Comput. Math. Appl. 58, 1457–1465 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, T.B., Deng, Z.S.: On the uniqueness of meromorphic functions that share three or two finite sets on annuli. Proc. Indian Acad. Sci. 122(2), 203–220 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fujimoto, H.: Uniqueness problem with truncated multiplicities in value distribution theory. Nagoya Math. J. 152, 131–152 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gundersen, G.: Meromorphic functions that share four values. Trans. Am. J. Math. 277(2), 545–567 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ishizaki, K., Toda, N.: Unicity theorems for meromorphic functions sharing four small functions. Kodai J. Math. 21, 350–371 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Khrystiyanyn, A.Y., Kondratyuk, A.A.: On the Nevanlinna theory for meromorphic functions on annuli, I. Math. Stud. 23(01), 19–30 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Khrystiyanyn, A.Y., Kondratyuk, A.A.: On the Nevanlinna theory for meromorphic functions on annuli. II. Math. Stud. 24(02), 57–68 (2005)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Li, X., Yi, H.-X., Hu, H.: Uniqueness results of meromorphic functions whose derivates share four small functions. Acta. Math. Sci. 32B, 1593–1606 (2012)zbMATHGoogle Scholar
  12. 12.
    Lund, M., Ye, Z.: Nevanlinna theory of meromorphic functions on annuli Sci. Chin. Math. 53, 547–554 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Nevanlinna, R.: Einige Eideutigkeitssäte in der theorie der meromorphen funktionen. Acta. Math. 48, 367–391 (1926)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Noguchi, J., Ochiai, T.: Introduction to Geometric Function Theory in Several Complex Variables. Trans. Math. Monogr. 80 Amer. Math. Soc., Providence, Rhode Island (1990)Google Scholar
  15. 15.
    Si, D.Q.: Unicity of meromorphic functions sharing some small function. Int. J. Math. 23(9) (2012).

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam
  2. 2.Thang Long Institute of Mathematics and Applied SciencesHanoiVietnam
  3. 3.Division of MathematicsBanking AcademyHanoiVietnam
  4. 4.Department of MathematicsUniversity of Hanoi Textile IndustryHanoiVietnam
  5. 5.Faculty of Fundamental SciencesElectric Power UniversityHanoiVietnam

Personalised recommendations