The winding number of Pf+1Pf+1 for polynomials PP and meromorphic extendibility of ff

Let ΔΔ be the open unit disc in CC. The paper deals with the following conjecture: if ff is a continuous function on bΔbΔ such that the change of argument of Pf+1Pf+1 around bΔbΔ is nonnegative for every polynomial PP such that Pf+1Pf+1 has no zero on bΔbΔ then ff extends holomorphically through ΔΔ. 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.