Meromorphic functions on annuli sharing finite sets with truncated multiplicity

The purpose of this article is twofold; first, 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; second, to show that if the polynomialPS(w)=(w−a1)⋯(w−aq) is a uniqueness polynomial for admissible meromorphic functions on an annulus A(R0) such that PS′(w) has exactly k distinct zeros and q>(5k+7)ℓ2ℓ−175, then the set S={a1,…,aq} is a finite range set with truncation level ℓ for admissible meromorphic functions on A(R0). This result extends the previous result on the finite range set (with truncation level ℓ=∞) for holomorphic functions on C of Fujimoto.