Prescribing inner parts of derivatives of inner functions

Let $\mathcal{J}$ be the set of inner functions whose derivatives lie in Nevanlinna class. In this note, we show that the natural map $F \to $\text{Inn}$(F’): \mathscr J/$\text{Aut}$($\mathbb{D}$) \to $\text{Inn}$/S^1$ is is injective but not surjective. More precisely, we show that that the image consists of all inner functions of the form $B{S}_{\mu }$ where $B$ is a Blaschke product and $S_{\mu }$ is the singular factor associated to a measure $\mu$ whose support is contained in a countable union of Beurling-Carleson sets. Our proof is based on extending the work of D. Kraus and O. Roth on maximal Blaschke products to allow for singular factors. This answers a question raised by K. Dyakonov.