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)].


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