On the cardinality of the unique range sets for meromorphic and entire functions (Q1572813)

By the Nevanlinna theory of meromorphic functions the authors obtain the following results which improve M. Reinders' recent results (to appear in the Proc. Intern. Workshop on Value Distr. Theory Appl.). 1) Let \(n\) be an integer \(\geq 15\) and \(c\) be a complex number \(\neq 0,1\), \(P(w)=(n-1)(n-2) w^n/2-n(n-2) w^{n-1}+ n(n-1)w^{n-2}/2-c\) and \(S=\{w\in \mathbb{C}: P(w)=0\}\). If \(f\) and \(g\) are two non-constant meromorphic functions satisfying \(E_1(f,S)= E_1(g, S)\), then \(f\equiv g\), where \(E_1(f,S)=\{\) the simple zeroes of \(f(z)-s_i,s_i\in S\}\) etc. 2) Let \(n\) be an integer \(\geq 9\) and \(c\) be a complex number \(\neq 0,1 \), \(P(w)\) and \(S\) be as in 1). If \(f\) and \(g\) are two non-constant meromorphic functions satisfying \(\Theta (\infty,f)\) and \(\Theta (\infty,g)>5/6\) and \(E_1(f, S)= E_1(g,S)\), then \(f\equiv g\), where \(\Theta (\cdot)\) is a usual notation in the Nevanlinna theory.