Revistes Catalanes amb Accés Obert (RACO)

Meromorphic extendibility and the argument principle

J. Globevnik


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 $\Delta$.

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