Geodesic
Finiteness of meromorphic functions on an annulus sharing four values regardless of multiplicity
Mathematica Bohemica, Tome 145 (2020) no. 2, pp. 163-176
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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 {\it 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.
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 {\it 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.
DOI : 10.21136/MB.2019.0121-17
Classification : 30D35, 32H30
Keywords: meromorphic function; Nevanlinna theory; annulus 
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Si, Duc Quang; Tran, An Hai. Finiteness of meromorphic functions on an annulus sharing four values regardless of multiplicity. Mathematica Bohemica, Tome 145 (2020) no. 2, pp. 163-176. doi: 10.21136/MB.2019.0121-17

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