Bloch-to-BMOA pullbacks on the disk

For a given holomorphic self map p of the unit disk, we consider the Bloch-to-BMOA composition property (pullback property) of Our results are (1) cannot have the pullback property if touches the boundary too smoothly, (2) while has the pullback property if touches the boundary rather sharply. One of these results yields an interesting consequence completely contrary to a higher dimensional result which has been known. These results resemble known results concerning the compactness of composition operators on the Hardy spaces. Some remarks in that direction are included.