Meromorphic extendibility and the argument principle

Abstract

Let ∆ be the open unit disc in C. Given a continuous function ϕ: b∆ → C\{0} denote by W(ϕ) the winding number of ϕ around the origin. We prove that a continuous function f : b∆ → C extends meromorphically through ∆ if and only if there is a number N ∈ N ∪ {0} such that W(Pf + Q) ≥ -N for every pair P, Q of polynomials such that Pf + Q 6= 0 on b∆. If this is the case then the meromorphic extension has at most N poles in ∆.