Meromorphic functions that share two sets
Let \(f\) be a meromorphic function in \(\mathbb{C}\) and \(S\) be a subset of \(\widehat \mathbb{C}\). We define by \(E_f(s): =\cup_{a\in S} \{z\in\mathbb{C} |f(z)- a=0\}\) where each zero of \(f-a\) will be counted with multiplicities. We denote by \(\overline E_f(s)\) the set of all distinct points of \(E_f(s)\). One says the meromorphic functions \(f\) and \(g\) share the set \(S\) \(CM\) if \(E_f(s) =E_g(s)\) and they share \(S\) \(IM\) if \(\overline E_f(s)= \overline E_g(s)\). With \(w:= \cos(2\pi/n) +i\sin (2\pi/n)\), \(n\in\mathbb{N}\), we define \(S_1:= \{1,w, \dots, w^{n-1}\}\) and denote by \(S_2:= \{\infty\}\) and \(S_3:=\{0\}\). Then the authors prove the following theorems: Let \(f\) and \(g\) be meromorphic in the plane. Suppose that \(f\) and \(g\) fulfill one of the following conditions (1) \(f\) and \(g\) share \(S_1 CM\) and \(S_2\) \(IM\) with \(n\geq 6\) or (2) \(f\) and \(g\) share \(S_1\) \(IM\) and \(S_2\) \(CM\) with \(n\geq 10\) or (3) \(f\) and \(g\) share \(S_1\) and \(S_2\) \(IM\) with \(n\geq 11\) then either \(f=tg\) or \(f\cdot g=t\) where \(t^n=1\). With these results they improve earlier results of F. Gross, C. F. Osgood, H.-X. Yi, G. D. Song, N. Li and others.
uniqueness theorems for meromorphic functions
scientific article; zbMATH DE number 1089816