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On the Boundary Dieudonné–Pick Lemma

Author

Listed:
  • Olga Kudryavtseva

    (Moscow Center of Fundamental and Applied Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Department of Applied Mathematics, Volgograd State Technical University, 400005 Volgograd, Russia)

  • Aleksei Solodov

    (Moscow Center of Fundamental and Applied Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia)

Abstract

The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

Suggested Citation

  • Olga Kudryavtseva & Aleksei Solodov, 2021. "On the Boundary Dieudonné–Pick Lemma," Mathematics, MDPI, vol. 9(10), pages 1-9, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1108-:d:554084
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