Meromorphic functions on annuli sharing finite sets with truncated multiplicity

The purpose of this paper has twofold. The first is to establish a second main theorem for meromorphic functions on annuli and meromorphic function targets (may not be small functions) with truncated counting functions (truncation level 1) and with a detailed estimate for the error term. The second is to show that if the polynomial $$P_S(w)=(w-a_1)\cdots (w-a_q)$$ is a uniqueness polynomial for admissible meromorphic functions on an annulus $\mathbb A(R_0)$ such that $P'_S(w)$ has exactly $k$ distinct zeros and $q>\frac{(5k+7)\ell}{2\ell-175}$, then the set $S=\{a_1,\ldots,a_q\}$ is a finite range set with truncation level $\ell$ for admissible meromorphic functions on $\mathbb A(R_0)$. This result extends the previous result on the finite range set (with truncation level $\ell=\infty$) for holomorphic functions on $\mathbb C$ of H. Fujimoto.