The Schwarz Lemma: Rigidity and Dynamics

Elin, Mark; Jacobzon, Fiana; Levenshtein, Marina; Shoikhet, David

doi:10.1007/978-3-319-01806-5_3

Cited by 27 publications

(7 citation statements)

References 103 publications

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“…These classical results characterize the behavior of the derivative on classes B[0] and B [1], respectively. The interest in developing these results stems from their applications in many directions, e.g., geometric function theory, hyperbolic geometry, complex dynamics, infinite-dimensional analysis, and the theory of composition operators (see [4][5][6][7][8]).…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

On the Boundary Dieudonné–Pick Lemma

Kudryavtseva

Солодов

2021

Mathematics

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“…Equality in these inequalities (in the first one, for z = 0) occurs only if f (z) = ze iθ , where θ is a real number ( [8], p. 329). For historical background about the Schwarz lemma and its applications on the boundary of the unit disc, we refer to (see [2,7]).…”

Section: Introductionmentioning

confidence: 99%

Some Estimates for Holomorphic Functions at the Boundary of the Unit Disc

Örnek¹

2020

KgJMat

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“…The study of holomorphic mappings of a disk into itself constitutes a significant part of geometric function theory (see, e.g., the papers [1,2] and the bibliography in them). We are interested in the distortion theorems for such mappings that take into account their boundary behavior [3].…”

Section: Introductionmentioning

confidence: 99%