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Mat. Zametki, 2017, Volume 102, Issue 4, Pages 559–564 (Mi mz11659)  

This article is cited in 2 scientific papers (total in 2 papers)

The Kraus Inequality for Multivalent Functions

V. N. Dubininab

a Far Eastern Federal University, Vladivostok
b Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Vladivostok

Abstract: For a holomorphic function $f,f'(0)\ne 0$, in the unit disk $U$, we establish a geometric constraint on the image $f(U)$ for which the classical Kraus inequality $|S_{f}(0)|\le 6$ holds; earlier, it was known only in the case of the conformal mapping of $f$. Here $S_{f}(0)$ is the Schwarzian derivative of the function $f$ calculated at the point $z=0$. The proof is based on the strengthened version of Lavrentev's theorem on the extremal decomposition of the Riemann sphere into two disjoint domains.

Keywords: Schwarzian derivative, holomorphic function, condenser capacity.

Funding Agency Grant Number
Russian Science Foundation 14-11-00022
This work was supported by the Russian Science Foundation under grant 14-11-00022.


DOI: https://doi.org/10.4213/mzm11659

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English version:
Mathematical Notes, 2017, 102:4, 516–520

Bibliographic databases:

Document Type: Article
UDC: 517.54
Received: 29.04.2017

Citation: V. N. Dubinin, “The Kraus Inequality for Multivalent Functions”, Mat. Zametki, 102:4 (2017), 559–564; Math. Notes, 102:4 (2017), 516–520

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. E. G. Prilepkina, “O $n$-garmonicheskom radiuse oblastei v $n$-mernom evklidovom prostranstve”, Dalnevost. matem. zhurn., 17:2 (2017), 246–256  mathnet  elib
    2. Dubinin V.N., “Two-Point Distortion Theorems and the Schwarzian Derivatives of Meromorphic Functions”, J. Math. Anal. Appl., 467:1 (2018), 371–378  crossref  mathscinet  zmath  isi  scopus
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