On uniqueness polynomials for meromorphic functions

Abstract

A polynomial $P(w)$ is called a uniqueness polynomial (or a uiqueness polynomial in a broad sense) if $P(f) = cP(g)$ (or $P(f) = P(g)$) implies $f = g$ for any nonzero constant $c$ and nonconstant meromorphic functions $f$ and $g$ on $\mathbf C$. We consider a monic polynomial $P(w)$ without multiple zero whose derivative has mutually distinct $k$ zeros $e_j$ with multiplicities $q_j$. Under the assumption that $P(e_\ell) \not= P(e_m)$ for all distinct $\ell$ and $m$, we prove that $P(w)$ is a uniqueness polynomial in a broad sense if and only if $\sum_{\ell>m}q_\ell q_m > \sum_{\ell} q_\ell$. We also give some sufficient conditions for uniqueness polynomials.