Meromorphic functions on annuli sharing finite sets with truncated multiplicity

TL;DR

This paper develops a second main theorem for meromorphic functions on annuli with truncated multiplicities and demonstrates conditions under which certain polynomial sets form finite range sets for these functions.

Contribution

It extends the second main theorem to annuli with truncated counting functions and generalizes finite range set results to meromorphic functions with specific polynomial conditions.

Findings

Established a second main theorem for meromorphic functions on annuli with truncated counting functions.

Proved that certain polynomial sets are finite range sets for admissible meromorphic functions on annuli.

Extended previous results from holomorphic functions on complex plane to meromorphic functions on annuli.

Abstract

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.