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Keywords = Julia–Carathéodory theorem

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9 pages, 244 KB  
Article
On the Boundary Dieudonné–Pick Lemma
by Olga Kudryavtseva and Aleksei Solodov
Mathematics 2021, 9(10), 1108; https://doi.org/10.3390/math9101108 - 13 May 2021
Cited by 4 | Viewed by 2727
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 [...] Read more.
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. Full article
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