The order of a linearly invariant family in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>

Chuaqui, Martin; Hernández, Rodrigo

doi:10.1016/j.jmaa.2012.08.049

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Cited by 3 publications

(2 citation statements)

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“…In the final section we estimate the order of the families treated. In [3], a generalization of Pommerenke's variational methods was employed to estimate the order of the families in B n , but linear invariance was required. Here instead, we appeal to a dilation argument that yields Lipschitz continuity for the order of slightly larger families.…”

Section: Introductionmentioning

confidence: 99%

Families of Homomorphic Mappings in the Polydisk

Chuaqui¹,

Hernández²

2022

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We study classes of locally biholomorphic mappings defined in the P n that have bounded Schwarzian operator in the Bergman metric. We establish important properties of specific solutions of the associated system of differential equations, and show a geometric connection between the order of the classes and a covering property. We show for modified and slightly larger classes that the order is Lipschitz continuous with respect to the bound on the Schwarzian, and use this to estimate the order of the original classes.

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Section: Introductionmentioning

confidence: 99%

Families of Homomorphic Mappings in the Polydisk

Chuaqui¹,

Hernández²

2022

Preprint

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“…In [7], we adapt the variational method introduced in [29] to estimate the order of the family F α in terms of α and the dimension n. As a consequence, we are able to provide, in terms of n, a range for the Schwarzian norm that ensures convexity, and later on, univalence and extensions.…”

Section: Introductionmentioning

confidence: 99%