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Let D be the open unit disc in C. The paper deals with the followingconjecture: If f is a continuous function on bD such that the change ofargument of Pf+1 around bD is nonnegative for every polynomial P such that Pf+1has no zero on bD then f extends holomorphically through D. We prove a relatedresult on meromorphic extendibility for smooth functions with finitely manyzeros of finite order, which, in particular, implies that the conjecture holdsfor real analytic functions.

The winding number of PF+1 for polynomials P and meromorphic extendibility of F