Free interpolation by nonvanishing analytic functions

We are concerned with interpolation problems in H-infinity where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence {z(j)} in the unit disk, we ask whether there exists a nontrivial minorant {epsilon(j)} ( i. e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem f(z(j)) = a(j) has a nonvanishing solution f is an element of H-infinity whenever 1 >= vertical bar a(j)vertical bar >= epsilon(j) for all j. The sequences {zj} with this property are completely characterized. Namely, we identify them as "thin" sequences, a class that arose earlier in Wolff's work on free interpolation in H-infinity boolean AND VMO.