## Publicacions Matemàtiques

### Meromorphic extendibility and the argument principle

Josip Globevnik

#### Abstract

Let $\Delta$ be the open unit disc in $\mathbb{C}$. Given a continuous function $\varphi \colon b\Delta \rightarrow \mathbb{C}\setminus \{ 0\}$ denote by $\mathcal{W} (\varphi )$ the winding number of $\varphi$ around the origin. We prove that a continuous function $f\colon b\Delta\rightarrow \mathbb{C}$ extends meromorphically through $\Delta$ if and only if there is a number $N\in \mathbb{N}\cup\{ 0\}$ such that $\mathcal{W} (Pf+Q)\geq -N$ for every pair $P$, $Q$ of polynomials such that $Pf+Q\not= 0$ on $b\Delta$. If this is the case then the meromorphic extension has at most $N$ poles in $\Delta$.

#### Article information

Source
Publ. Mat., Volume 52, Number 1 (2008), 171-188.

Dates
First available in Project Euclid: 17 December 2007

https://projecteuclid.org/euclid.pm/1197908701

Mathematical Reviews number (MathSciNet)
MR2384845

Zentralblatt MATH identifier
1149.30031

Subjects