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Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2016, Volume 158, Book 2, Pages 172–179 (Mi uzku1360)  

This article is cited in 1 scientific paper (total in 1 paper)

Estimates of the hyperbolic radius gradient and Schwarz–Pick inequalities for the eccentric annulus

D. Kh. Giniyatova

Kazan Federal University, Kazan, 420008 Russia

Abstract: Let $\Omega$ and $\Pi$ be hyperbolic domains in the complex plane $\mathbb C$. By $A(\Omega,\Pi)$ we shall designate the class of functions $f$ which are holomorphic or meromorphic in $\Omega$ and such that $f(\Omega)\subset\Pi$. Estimates of the higher derivatives $|f^{(n)}(z)|$ of the analytic functions from the class $A(\Omega,\Pi)$ with the punishing factor $C_n(\Omega,\Pi)$ is one of the main problems of geometric theory of functions. These estimates are commonly referred to as Schwarz–Pick inequalities. Many results concerning this problem have been obtained for simply connected domains. Therefore, the research interest in such problems for finitely connected domains is natural. As known, the constant $C_2(\Omega,\Pi)$ for any pairs of hyperbolic domains depends only on the hyperbolic radius gradient of the corresponding domains. The main result of this paper is estimates of the hyperbolic radius gradient and the punishing factor in the Schwarz–Pick inequality for the eccentric annulus. We also consider the extreme case – the randomly punctured circle.

Keywords: Poincare metrics, Schwarz–Pick inequalities, conformal mapping, punishing factors.

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Bibliographic databases:
UDC: 517.544
Received: 02.10.2015

Citation: D. Kh. Giniyatova, “Estimates of the hyperbolic radius gradient and Schwarz–Pick inequalities for the eccentric annulus”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 158, no. 2, Kazan University, Kazan, 2016, 172–179

Citation in format AMSBIB
\Bibitem{Gin16}
\by D.~Kh.~Giniyatova
\paper Estimates of the hyperbolic radius gradient and Schwarz--Pick inequalities for the eccentric annulus
\serial Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
\yr 2016
\vol 158
\issue 2
\pages 172--179
\publ Kazan University
\publaddr Kazan
\mathnet{http://mi.mathnet.ru/uzku1360}
\elib{https://elibrary.ru/item.asp?id=26416796}


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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. D. Kh. Giniyatova, “Tochnyi poryadok rosta mazhoranty v neravenstve Shvartsa–Pika dlya zhestkosti krucheniya”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 6(37), 18–27  mathnet  crossref
  • Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki
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