Prescribing inner parts of derivatives of inner functions

@article{Ivrii2019PrescribingIP,
  title={Prescribing inner parts of derivatives of inner functions},
  author={O. V. Ivrii},
  journal={Journal d'Analyse Math{\'e}matique},
  year={2019}
}
  • O. Ivrii
  • Published 31 January 2017
  • Mathematics
  • Journal d'Analyse Mathématique
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… 
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