Meromorphic Functions Having the Same Inverse Images of Four Values on Annuli
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.
KeywordsMeromorphic function Nevanlinna theory Annulus
Mathematics Subject ClassificationPrimary 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.
- 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.Si, D.Q.: Unicity of meromorphic functions sharing some small function. Int. J. Math. 23(9) (2012). https://doi.org/10.1142/S0129167X12500887