## 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

entire functions

Zbl 0357.30007
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### 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
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