On a result of H. Fujimoto

Abstract

Let $P(\omega)$ be a uniqueness polynomial of degree $q$ without multiple zeros whose derivative has mutually distinct $k$ zeros $d_l$ with multiplicities $q_l$ for $l=1, 2, \ldots, k$ respectively, and let $S:=\{a_1, a_2, \cdots, a_q\}$ be the zero set of $P(\omega)$. Under the assumption that $P(d_{l_s})\neq P(d_{l_t})$ $(1\leq l_s < l_t\leq k)$, we give some sufficient conditions for the set $S$ to be a unique range set with some weak value-sharing hypothesis, namely, to satisfy the condition that $\sum_{j=1}^q\nu_{f,m_0)}^{a_j}\equiv\sum_{j=1}^q\nu_{g,m_0)}^{a_j}$ ($m_0\in\mathbb{Z}^+\cup\{\infty\}$) implies $f\equiv g$ for any two nonconstant meromorphic or entire functions $f$ and $g$ on $\mathbb{C}$, which improve a result of H. Fujimoto. Also, we discuss some other related topics.