Meromorphic solutions of certain functional equations (Q652091)

Let \(P(z)\) be a polynomial. By means of the Nevanlinna theory, the authors consider meromorphic solutions of the functional equation \[ P(f)f'P(g)g'=1.\tag{1} \] They assume that \(P(z)\) has at least two distinct zeros. This research is in the spirit of [\textit{C.-C. Yang} and \textit{X. Hua}, Ann. Acad. Sci. Fenn., Math. 22, No. 2, 305--406 (1997; Zbl 0890.30019)], in which the functional equation \(f^nf'g^ng'=1\) was considered. It is proved in the paper under the review that if \(P\) has three distinct zeros, then (1) has no meromorphic solutions \(f\) and \(g\). As a corollary, the authors give a uniqueness result. For the case when \(P\) has just two distinct zeros, they construct a functional equation of the form (1) that possesses transcendental meromorphic solutions \(f\) and \(g\) using the \(\wp\)-function.