The range set of meromorphic derivatives. (English) Zbl 0934.30027

Suppose \(f\) is a nonconstant meromorphic function defined on the complex plane \(\mathbb{C}\), \(S\) is a set of distinct complex numbers, and \(E_f(S)= \bigcup_{a\in S}\{z\mid f(z)- a= 0\}\), where a zero of \(f-a\) of multiplicity \(m\) is repeated \(m\) times in \(E_f(S)\). The author’s main result follows.
Assume \(S= \{z\mid z^n+ az^{n- 1}+ b=0\}\), where \(a\), \(b\) are nonzero constants such that \(z^n+ az^{n-1}+ b=0\) has no repeated root, \(n\) is a positive integer greater than 7, and \(f\) and \(g\) are meromorphic functions for which \(f'\) and \(g'\) are nonconstant. If \(E_{f'}(S)= E_{g'}(S)\), \(E_f(\{\infty\})= E_g(\{\infty\})\), and \(f(z_0)= g(z_0)\neq \infty\) for some \(z_0\in\mathbb{C}\), then \(f\equiv g\). The paper relies on considerable technical expertise in the use of the Nevanlinna calculus.
Reviewer: L.R.Sons (DeKalb)


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