The winding number of for polynomials and meromorphic extendibility of

Let Δ be the open unit disc in C . The paper deals with the following conjecture: if f is a continuous function on b Δ such that the change of argument of P f + 1 around b Δ is nonnegative for every polynomial P such that P f + 1 has no zero on b Δ then f extends holomorphically through Δ . 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.