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 Quang
  • 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.


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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

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