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 .