Strengthening of Landau’s Theorem     for Holomorphic Self-Mappings of a Disk     with Fixed Points

Солодов, Алексей Петрович

doi:10.1134/s0001434620090382

Cited by 11 publications

(2 citation statements)

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“…However, the sharpness of inequality (11) was not discussed. In Section 5, we give another proof of inequality (11) based on the Julia-Carathéodory theorem and Pick's ideas. We also establish its nonimprovability and describe the class of the extremal functions.…”

Section: Boundary Dieudonné-pick Lemma and Its Consequencesmentioning

confidence: 99%

“…Note that classes of functions with several fixed points are studied in connection with various problems, e.g., the problem of fractional iteration of holomorphic function [9], the problem of finding domains of univalence [10][11][12], the problem of describing the Taylor coefficients [13,14], as well as estimating the angular derivatives [15] and the Schwarzian derivatives [16].…”

Section: Introductionmentioning

confidence: 99%

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On the Boundary Dieudonné–Pick Lemma

Kudryavtseva

Солодов

2021

Mathematics

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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.

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Section: Boundary Dieudonné-pick Lemma and Its Consequencesmentioning

confidence: 99%