×

Unicity theorems for meromorphic or entire functions. (English) Zbl 0809.30024

Let \(S_ 1 = \{1,w,w^ 2, \dots, w^{n-1}\}\), \(S_ 2 = \{a,b\}\), \(S_ 3 = \{0\}\) and \(S_ 4 = \{\infty\}\) and where \(ab \neq 0\), \(a^ n \neq b^ n\), \(a^{2n} \neq 1\), \(b^{2n} \neq 1\), \(a^ n b^ n \neq 1\) and \(w = \exp (2 \pi i/n)\) \((n>6)\) and \(\in \mathbb{N})\). For any set \(S\), let \(E_ f(S) = f^{-1} (S)\) where we take due account of multiplicity. The author proves that 1) Suppose that \(f\) and \(g\) are nonconstant meromorphic functions satisfying \(E_ f(S_ j) = E_ g\) \((S_ j)\) \((j=1,2,3)\). Then \(f=g\). 2) Suppose that \(f\) and \(g\) are nonconstant meromorphic functions satisfying \(E_ f(S_ j) = E_ g\) \((S_ j)\) \((j=1,2,4)\). Then \(f=g\). 3) Suppose that \(f\) and \(g\) are nonconstant entire functions satisfying \(E_ f(S_ j) = E_ g(S_ j)\) \((j=1,2)\). Then \(f=g\). 4) Suppose that \(f\) and \(g\) are nonconstant meromorphic functions satisfying \(E_ f(S_ j) = E_ g\) \((S_ j)\) \((j=1,4)\). Then either \(f=cg\) \((c^ n=1)\), or \(fg=d\) \((d^ n=1)\). The 3) answers the question posed by F. Gross [Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes Math. Vol. 599, 51-67 (1977; Zbl 0357.30007)].

MSC:

30D30 Meromorphic functions of one complex variable (general theory)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Citations:

Zbl 0357.30007
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.2307/1999223 · Zbl 0508.30029
[2] DOI: 10.1007/BFb0096825
[3] Gross, Factorization of meromorphic functions (1972) · Zbl 0266.30006
[4] DOI: 10.2307/1994690 · Zbl 0157.12903
[5] Hayman, Meromorphic functions (1964)
[6] DOI: 10.1080/17476939008814415 · Zbl 0701.30025
[7] Yi, On a result of Gross (1990) · Zbl 0714.30028
[8] Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes (1929)
[9] DOI: 10.2996/kmj/1138039280 · Zbl 0712.30029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.