# Prescribing inner parts of derivatives of inner functions

## TL;DR

This paper characterizes the inner functions that can be obtained as derivatives of certain inner functions with derivatives in the Nevanlinna class, showing the map's injectivity and describing its image explicitly.

## Contribution

It extends previous work on maximal Blaschke products to include singular factors, providing a complete description of the image of the natural map.

## Key findings

- The map from inner functions to their derivatives' inner parts is injective.
- The image consists of inner functions formed by Blaschke products times singular factors with support in countable unions of Beurling-Carleson sets.
- The work answers a question posed by K. Dyakonov about the prescribed inner parts of derivatives.

## Abstract

Let $\mathscr J$ 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 $BS_\mu$ 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.