## Some further results on meromorphic functions that share two sets.(English)Zbl 1064.30023

Let $$f(z)$$ be a meromorphic function in the complex plane. For $$a\in \mathbb{C}$$, the authors defined $$E(a,f)$$ the set of zeros of $$f(z)-a$$ with counting multiplicity. For $$\infty$$, $$E(\infty,f)$$ denotes the set of poles of $$f(z)$$ counting multiplicity. Further, they consider the set $$E_{m)}(a,f)$$ the set of zeros of $$f(z)- a$$ with multiplicity less than or equal to $$m$$, where $$m$$ is a positive integer. The sets $$E_{m)}(\infty,f)$$ can be defined in a similar manner. Let $$S$$ be the subset of distinct elements in $$\mathbb{C}\cup\{\infty\}$$. Define $$E(S,f)= \bigcup_{a\in S} E(a,f)$$ and $$E_{m)}(S, f)= \bigcup_{a\in S} E_{m)}(a,f)$$. Let $$f(z)$$ and $$g(z)$$ be two nonconstant meromorphic functions. If $$E(S,f)= E(S,g)$$, it is said that $$f(z)$$ and $$g(z)$$ share the set $$S$$ CM. In particular, in case $$S= \{a\}$$, $$a\in\mathbb{C}\cup\{\infty\}$$, we call that $$f(z)$$ and $$g(z)$$ share value $$a$$ CM if $$E(a,f)= E(a,g)$$. Results on the sharing value problems can be found e.g., G. G. Gundersen [Complex Variables, Theory Appl. 20, No. 1–4, 99–106 (1992; Zbl 0773.30032)], and C.-C. Yang and H.-X. Yi [Theory of the uniqueness of meromorphic function, Beijing Science Press, 1995]. The authors consider the problem on sharing two sets. There are some results on this problem. For example, P. Li and C.-C. Yang [On the unique range set of meromorphic functions, Proc. Am. Math. Soc. 124, No. 1, 177–185 (1996; Zbl 0845.30018)] proved that there exists a set with 15 elements such that any two functions $$f(z)$$ and $$g(z)$$ satisfying $$E(S,f)= E(S,g)$$ and $$E(\infty,f)= E(\infty,g)$$ must be identical. The authors’ key idea is to consider the polynomial below, and the set $$S_P= \{w\mid P(w)= 0\}$$ $P(w)= aw^n- n(n- 1)w^2+ 2n(n- 2)bw- (n-1)(n- 2)b^2,$ where $$n\geq 8$$ is an integer, and $$a$$ and $$b$$ are two nonzero finite complex numbers such that $$ab^{n-2}\neq 2$$. It is shown that for $$m> 1$$, $$m+ n\geq 11$$, two functions $$f(z)$$ and $$g(z)$$ satisfying $$E_{m)}(S_P,f)= E_m(S_P, g)$$ and $$E(\infty,f)= E(\infty,g)$$ must be identical. This result is an improvement for the result due to the one of the authors H. Yi [Acta Math. Sin. 45, No. 1, 75–82 (2002; Zbl 1092.30051)].

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

### Keywords:

shared set; uniqueness meromorphic function

### Citations:

Zbl 0773.30032; Zbl 0845.30018; Zbl 1092.30051