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

Yi, Hongxun

J. Shandong Univ., Nat. Sci. Ed. 29, No. 3, 241-246 (1994).

[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 \(S_3= \{\infty \}\).

Reviewer: A.F.Grishin (Khar’kov)

MSC:

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

Keywords:

finite set; unqiueness theorems; meromorphic functions; entire functions

Citations:

Zbl 0821.30024