The author considers questions which are closely related to uniqueness polynomials and uniqueness range sets for meromorphic functions. Firstly, he gives conditions for a polynomial \(P\) such that \(P(f)= cP(g)\) for any nonzero constant \(c\) and nonconstant meromorphic functions \(f\) and \(g\) implies \(f= g\).
Next, he shows some conditions on a finite set \(S\) such that \(S\) is a uniqueness range set for meromorphic functions. For instance he gives a condition in case of \(\text{card }S> 20\).