Algebras generated by the disc algebra and bounded harmonic functions

Let $D$ denote the open unit disc, and let $A(D)$ denote the disc algebra. The subsets $E$ of $\partial D$ such that the inclusion $A(D)[f,{\overline f}]\supset C(\overline D)$ holds for every nonconstant $f\in H^\infty (D)$ continuous on $E$, or the inclusion $A(D)[f] \supset C(\overline D)$ holds for every bounded harmonic nonholomorphic function $f$ on $D$ continuous on $E$, are characterized. In the first case the condition is that $E$ has positive measure, and in the second case that $E$ has full measure in $\partial D$.