On the zero-one-pole set of a meromorphic function (Q1120026)

Let \(\{a_ n\}\), \(\{b_ n\}\) and \(\{p_ n\}\) be three disjoint sequences with no finite limit points. If it is possible to construct a meromorphic function f in the plane C whose zeros, one points and poles are exactly \(\{a_ n\}\), \(\{b_ n\}\) and \(\{p_ n\}\) respectively, where their multiplicities are taken into consideration, then the given tria \((\{a_ n\},\{b_ n\},\{p_ n\})\) is called a zero-one-pole set. The author proves several theorems related to zero-one-pole sets. A typical result follows: Theorem 1. Suppose that \((\{a_ n\},\{b_ n\},\{p_ n\})\), \((\{a_ n\}\cup \{b_ n\}\cup \{p_ n\})\neq \emptyset\) is a zero-one-pole set which is not unique. Let \(\{c_ n\}\), \(\{d_ n\}\) and \(\{q_ n\}\) be subsequences of \(\{a_ n\}\), \(\{b_ n\}\) and \(\{p_ n\}\) respectively such that \(\{c_ n\}\cup \{d_ n\}\cup \{q_ n\}\neq \emptyset\) and such that \[ \sum_{c_ n\neq 0}| c_ n|^{-1}+\sum_{d_ n\neq 0}| d_ n|^{-1}+\sum_{q_ n\neq 0}/q_ n|^{- 1}<+\infty. \] Then \((\{a_ n\}\setminus \{c_ n\}\), \(\{b_ n\}\setminus \{d_ n\}\), \(\{p_ n\}\setminus \{q_ n\}\) is not a zero- one-pole set of any nonconstant meromorphic function.