## Uniqueness of meromorphic functions and a question of Gross. II.(Chinese. English summary)Zbl 0828.30014

[For part I see the author in Sci. China, Ser. A 37, No. 7, 802-813 (1994; Zbl 0821.30024).]
The author proves new unqiueness theorems for entire and meromorphic functions. The origin is the work of R. Nevanlinna, 1929. Some of the theorems answer a question posed by F. Gross (1977). Let $$f$$ be a meromorphic function in $$\mathbb{C}$$, let $$S\subset \mathbb{C}$$, $$E_f (S)= \bigcup_{\alpha\in s} \{z$$: $$f(z)- \alpha= 0\}$$, where the zeros of the equation $$f(z)- \alpha=0$$ are counted with correct multiplicity. Let $$W= \exp ({{2\pi i}\over n})$$, $$u= \exp ({{2\pi i} \over m})$$, $$n,m\in \mathbb{N}$$. The sets $$S_j$$, $$j=1, 2, \dots, q$$ are given. Let $$f$$ and $$g$$ be meromorphic or entire functions. In which cases follow from the relations $$E_f (S_j)= E_g (S_j)$$, $$j=1, 2,\dots, q$$ that $$f\equiv g$$? The author gives a positive answer in the following cases:
1) $$S_1= \{a+b, a+bW, \dots, a+b W^{n-1}\}$$, $$S_2= \{c_1, \dots, c_p\}$$, $$n>8$$, $$p>1$$, $$b\neq 0$$, $$c_1\neq a$$, $$c_2\neq a$$, $$(c_k- a)^n \neq (c_j- a)^n$$ if $$k\neq j$$, $$(c_k- a)^n (c_j- a)^n\neq b^{2n}$$, $$f$$ and $$g$$ are meromorphic functions.
2) $$S_1$$ and $$S_2$$ are the same as in 1), $$n>4$$, $$p>0$$, assumption $$c_2\neq a_2$$ is omitted, $$f$$ and $$g$$ are entire functions.
3) $$S_1= \{a_1+ b_1, \dots, a_1+ b_1 w^{n-1} \}$$, $$S_2= \{a_2+ b_2, \dots, a_2+ b_2 u^{m-1} \}$$, $$n>8$$, $$m>8$$, $$a_1\neq a_2$$, $$b_1\neq 0$$, $$b_2\neq 0$$, $$f$$ and $$g$$ are meromorphic functions.
4) The assumptions are the same as in 3) but $$n>4$$, $$m>4$$, $$f$$ and $$g$$ are entire functions.
5) $$S_1= \{a+ b_1, \dots, a+ b_1 W^{n-1}\}$$, $$S_2= \{a+ b_2, \dots, a+ b_2 u^{m-1} \}$$, $$n>8$$, $$m>8$$, $$n$$ and $$m$$ are relatively prime, $$b_1^{2mn} \neq b_2^{2mn}$$, $$f$$ and $$g$$ are meromorphic functions.
6) The assumptions are the same as in 5) but $$n>4$$, $$m>4$$, $$f$$ and $$g$$ are entire functions.
Also, the author proves that the inequality $$n>8$$ in 1) and inequalities $$n,m>s$$ in 3) and 5) may be changed to inequalities $$n>6$$, $$n,m>6$$ if to add and the third set $$S_3= \{\infty \}$$.

### MSC:

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

Zbl 0821.30024