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On L 1-estimates of derivatives of univalent rational functions

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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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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    Anton D. Baranov & Konstantin Yu. Fedorovskiy

  2. National Research University Higher School of Economics, St. Petersburg, Russia

    Anton D. Baranov

  3. Bauman Moscow State Technical University, Moscow, Russia

    Konstantin Yu. Fedorovskiy

Authors
  1. Anton D. Baranov
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  2. Konstantin Yu. Fedorovskiy
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Corresponding author

Correspondence to Konstantin Yu. Fedorovskiy.

Additional information

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

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