Prescribing inner parts of derivatives of inner functions

2017/01/31 by Ivrii, Oleg · 1 citation
#30F45 #30J05 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1702.00090

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 → Inn(F'): \mathscr J/Aut(\mathbbD) → Inn/S¹ is is injective but not surjective. More precisely, we show that that the image consists of all inner functions of the form BS_μ where B is a Blaschke product and S_μ is the singular factor associated to a measure μ 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.

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