Drop of the smoothness of an outer function compared to the smoothness of its modulus, under restrictions on the size of boundary values

Let $F$ be an outer function on the unit disk. It is well known that its smootheness properties may be two times worse that those of the modulus of its boundary values, but under some restrictions on $\log|F|$ this gap becomes smaller. It is shown that the smoothness decay admits a convenient description in terms of a rearrangement invariant Banach function space containing $\log|F|$. All results are of pointwise nature.