Mathematica Bohemica, Vol. 145, No. 2, pp. 163-176, 2020
Finiteness of meromorphic functions on an annulus sharing four values regardless of multiplicity
Duc Quang Si, An Hai Tran
Received October 27, 2017. Published online June 10, 2019.
Abstract: This paper deals with the finiteness problem of meromorphic funtions on an annulus sharing four values regardless of multiplicity. We prove that if three admissible meromorphic functions $f_1$, $f_2$, $f_3$ on an annulus $\mathbb A({R_0})$ share four distinct values regardless of multiplicity and have the complete identity set of positive counting function, then $f_1=\nobreak f_2$ or $f_2=f_3$ or $f_3=f_1$. This result deduces that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicity truncated to level $2$ and sharing other three values regardless of multiplicity. This result also implies that there are at most three admissible meromorphic functions on an annulus sharing four values regardless of multiplicities. These results are a generalization and improvement of the previous results on finiteness problem of meromorphic functions on $\mathbb C$ sharing four values.
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Affiliations: Duc Quang Si, Department of Mathematics, Hanoi National University of Education, 136-Xuan Thuy, Cau Giay, 123106 Hanoi, Vietnam, and Thang Long Institute of Mathematics and Applied Sciences, Nghiem Xuan Yem, Hoang Mai, 100000 Hanoi, Vietnam, e-mail: quangsd@hnue.edu.vn; An Hai Tran, Division of Mathematics, Banking Academy, 12-Chua Boc, Dong Da, 100000 Hanoi, Vietnam, e-mail: trananhai@wru.vn
