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On L 1-estimates of derivatives of univalent rational functions

Abstract

We study the growth of the quantity ∫T|R′(z)|dm(z) for rational functions R of degree n which are bounded and univalent in the unit disk and prove that this quantity can grow like n γ, γ > 0, as n → ∞. Some applications of this result to problems of regularity of boundaries of Nevanlinna domains are considered. We also discuss a related result of Dolzhenko, which applies to general (non-univalent) rational functions.

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Correspondence to Konstantin Yu. Fedorovskiy.

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This research was partially supported by the Russian Foundation for Basic Research, grants 15-01-07531 and 16-01-00674, by the Ministry of Education and Science of the Russian Federation, projects 1.3843.2017 and 1.517.2016, and by Dmitry Zimin’s “Dynasty” Foundation.

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Baranov, A.D., Fedorovskiy, K.Y. On L 1-estimates of derivatives of univalent rational functions. JAMA 132, 63–80 (2017). https://doi.org/10.1007/s11854-017-0010-y

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  • DOI: https://doi.org/10.1007/s11854-017-0010-y