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Boundary Distortion and the Schwarzian Derivative of a Univalent Function in a Circular Annulus

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Abstract

New distortion theorems are proved for holomorphic univalent functions bounded in a circular annulus and preserving one of its boundary components. In particular, inequalities including the Schwarzian derivative at a boundary point of the annulus are established. All results follow from the properties of the conformal capacity of condensers and symmetrization.

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Notes

  1. See, e.g., [7, Sec. 2.1].

  2. Inequality (1) can also be obtained by using inequality (11) in [6] with a description of all cases of equality.

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Funding

This research was supported by the Mathematical Center in Akademgorodok under agreement no. 075-15-2022-281 of 05.04.2022 with the Ministry of Science and Higher Education of the Russian Federation.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences, Vladivostok, 690041, Russia

    V. N. Dubinin

  2. Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Novosibirsk, 630090, Russia

    V. N. Dubinin

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  1. V. N. Dubinin
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Correspondence to V. N. Dubinin.

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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 827–835 https://doi.org/10.4213/mzm13808.

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Dubinin, V.N. Boundary Distortion and the Schwarzian Derivative of a Univalent Function in a Circular Annulus. Math Notes 113, 776–783 (2023). https://doi.org/10.1134/S000143462305019X

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