Meromorphic extendibility and rigidity of interpolation

Let T be the unit circle, f be an α-Hölder continuous function on T, α>1/2, and A be the algebra of continuous function in the closed unit disk D¯ that are holomorphic in D. Then f extends to a meromorphic function in D with at most m poles if and only if the winding number of f+h on T is bigger or equal to −m for any h∈A such that f+h≠0 on T.