Unicity theorems for meromorphic functions (Q5970585)

Let \(f\) be a non-constant meromorphic function in the complex plane. The following two theorems will be proved in this paper: (1) Given a non-constant polynomial, if \(f,f'\) share zero, counting multiplicity (CM) and if \(f'(z)=a(z)\) always implies \(f(z)=a(z)\), then \(f=f'\). (2) Given a non-zero constant \(a\), if \(f,f'\) share zero CM and if \(f'(z)=a\) always implies \(f(z)=a\), then either \(f=f'\) or \(f(z)=2a/(1-ce^{-2z})\), where \(c\neq0\) is a constant. As a corollary of the first theorem, \(f=f'\) provided \(ff'\neq0\) and \(f,f'\) have the same fixed points. The above theorems are slight generalizations of previous results due to \textit{G. G. Gundersen} [J. Math. Anal. Appl. 75, 441--446 (1980; Zbl 0447.30018)] and \textit{Q. C. Zhang} [Acta Math. Sin. 45, No. 5, 871--876 (2002; Zbl 1103.30302)].