Meromorphic functions and their derivatives share two finite sets
Let \(n\) and \(k\) be two positive integers. H. X. Yi [cf., \textit{H. X. Yi} and \textit{C. C. Yang}, Uniqueness theory of meromorphic functions (2003; Zbl 1070.30011)] proved the following result: If two nonconstant entire functions \(f\) and \(g\) in \(\mathbb{C}\) share the set \(S_n=\{z\in \mathbb {C}: z^n-1=0\}\) and a value \(a\in\mathbb{C}-\{0\}\) with \(a^2\not\in S_n\) counting multiplicity, then \(f=g\) if \(n>4\). NEWLINENEWLINE\textit{M. Fang} [Bull. Malays. Math. Sci. Soc. (2) 24, No. 1, 7--16 (2001; Zbl 1078.30024)] obtained that if two nonconstant entire functions \(f\) and \(g\) in \(\mathbb{C}\) share the set \(S_n\) counting multiplicity, and if \(f^{(k)}\) and \(g^{(k)}\) share the set \(T=\{a,b,c\}\) counting multiplicity, where \(a\), \(b\) and \(c\) are distinct values in \(\mathbb{C}-\{0\}\) satisfyingNEWLINE\[NEWLINE a^2\not=bc,\qquad b^2\not=ac,\qquad c^2\not=ab,\tag{abc}NEWLINE\]NEWLINE then \(f=g\) if \(n>4\). NEWLINENEWLINEThe author proves that if two non-polynomial meromorphic functions \(f\) and \(g\) in \(\mathbb{C}\) share the set \(S_n\) and the value \(\infty\) counting multiplicity, and if \(f^{(k)}\) and \(g^{(k)}\) share the set \(T\) counting multiplicity, then \(f=g\) if \(n>5\). If the condition (abc) is cancelled, some relations of \(f\) and \(g\) are also given by the author.
scientific article; zbMATH DE number 1719571