Bloch-to-BMOA pullbacks on the disk

Abstract

For a given holomorphic self map $\varphi$ of the unit disk, we consider the Bloch-to-$BMOA$ composition property (pullback property) of $\varphi$. Our results are $(1)$ $\varphi$ cannot have the pullback property if $\varphi$ touches the boundary too smoothly, $(2)$ while $\varphi$ has the pullback property if $\varphi$ 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.