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Sibirskii Matematicheskii Zhurnal, 2010, Volume 51, Number 2, Pages 285–302 (Mi smj2083)  

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

Distortion theorems for univalent meromorphic functions on an annulus

V. N. Dubinin, E. G. Prilepkina

Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences
Full-text PDF (398 kB) Citations (2)
References:
Abstract: We apply the capacity and symmetrization methods to distortion theorems for analytic functions in an annulus. We show that the classical Teichmüller estimate for the capacity of a doubly-connected domain yields a series of the already known and new inequalities for univalent functions. In particular, we supplement the results of Grötzsch, Duren, and Huckemann. Using the dissymmetrization of condensers we establish sharp estimates for local distortion and the distortion of level curves in $n\ge2$ symmetric directions. In terms of Robin functions we give an analog of the Nehari inequality: some general distortion theorem for several points taking into account the boundary behavior of the function and describing the cases of equalities. As a corollary we give analogs of some inequalities of Solynin, Pommerenke, and Vasil'ev that were obtained previously for univalent and bounded functions in a disk. We prove a distortion theorem that involves the Schwarzian derivatives at symmetric points on the unit circle.
Keywords: meromorphic function, univalent function, distortion theorem, Schwarzian derivative, doubly-connected domain, annulus, condenser capacity, dissymmetrization, Green's function, Robin function.
Received: 27.01.2009
English version:
Siberian Mathematical Journal, 2010, Volume 51, Issue 2, Pages 229–243
DOI: https://doi.org/10.1007/s11202-010-0023-6
Bibliographic databases:
Document Type: Article
UDC: 517.54
Language: Russian
Citation: V. N. Dubinin, E. G. Prilepkina, “Distortion theorems for univalent meromorphic functions on an annulus”, Sibirsk. Mat. Zh., 51:2 (2010), 285–302; Siberian Math. J., 51:2 (2010), 229–243
Citation in format AMSBIB
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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