Free interpolation by nonvanishing analytic functions

We are concerned with interpolation problems in $H^\infty$ 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 $\{\varepsilon_j\}$ (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 $f\in H^\infty$ whenever $1\ge\vert a_j\vert\ge\varepsilon_j$ for all $j$. The sequences $\{z_j\}$ 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 $H^\infty\cap$VMO.