Complex meromorphic functions \(f^{\prime}P^{\prime}(f), g^{\prime}P^{\prime}(g)\) sharing a small function
Let \(f\) and \(g\) be transcendental meromorphic functions in \(\mathbb C\), let \(\alpha\) be a small meromorphic function with respect to both \(f\) and \(g\), and let \(P\) be a polynomial of uniqueness for meromorphic functions in \(\mathbb C\). Five theorems with multiple corollaries of the following form are given: if \(f'P'(f)\) and \(g'P'(g)\) share \(\alpha\) counting multiplicity, and the multiplicity orders of the zeros of \(P'\) satisfy certain inequalities, then \(f=g\). The results extend or complement theorems in [\textit{W. Lin} and \textit{H. Yi}, Complex Variables, Theory Appl. 49, No. 11, 793--806 (2004; Zbl 1067.30065)] and [\textit{K. Boussaf} et al., Bull. Sci. Math. 136, No. 2, 172--200 (2012; Zbl 1255.30047)].
scientific article; zbMATH DE number 6121457