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

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