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Ufimsk. Mat. Zh., 2019, Volume 11, Issue 2, Pages 3–18 (Mi ufa468)  

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

Conformal invariants of hyperbolic planar domains

F. G. Avkhadiev, R. G. Nasibullin, I. K. Shafigullin

Kazan Federal University, Lobachevsky Institute of Mathematics and Mechanics, Kremlevskaya str., 18, 420008, Kazan, Russia

Abstract: We consider planar hyperbolic domains and conformally invariant functionals defined as sharp constants for Hardy type inequalities. We study relationships between these functionals and optimal constants in hyperbolic isoperimetric inequalities. The studied Hardy type inequalities involve weight functions depending on a hyperbolic radius of a domain and are conformally invariant. We prove that the positivity of Hardy constants is connected with existence of some hyperbolic isoperimetric inequalities of a special kind. We also prove a comparison theorem for Hardy constants with different numerical parameters and we study the relationships between the linear hyperbolic isoperimetric inequality in a domain and Euclidean maximum modulus of this domain. In the proofs, an essential role is played by characteristics of domains with uniformly perfect boundary. In addition, we generalize certain results from the papers J.L. Fernández, J.M. Rodríguez, “The exponent of convergence of Riemann surfaces, bass Riemann surfaces”, Ann. Acad. Sci. Fenn. Series A. I. Mathematica. 15, 165–183 (1990); V. Alvarez, D. Pestana, J.M. Rodríguez, “Isoperimetric inequalities in Riemann surfaces of infinite type”, Revista Matemática Iberoamericana, 15:2, 353–425 (1999).

Keywords: Poincaré metric, hyperbolic isoperimetric inequality, uniformly perfect set, Hardy type inequality.

Funding Agency Grant Number
Russian Science Foundation 18-11-00115
The reported study was funded by RFBR according to the research project no. 18-11-00115.

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English version:
Ufa Mathematical Journal, 2019, 11:2, 3–18 (PDF, 413 kB); https://doi.org/10.13108/2019-11-2-3

Bibliographic databases:

UDC: 517.518
MSC: 30F45, 30A10
Received: 20.02.2019

Citation: F. G. Avkhadiev, R. G. Nasibullin, I. K. Shafigullin, “Conformal invariants of hyperbolic planar domains”, Ufimsk. Mat. Zh., 11:2 (2019), 3–18; Ufa Math. J., 11:2 (2019), 3–18

Citation in format AMSBIB
\by F.~G.~Avkhadiev, R.~G.~Nasibullin, I.~K.~Shafigullin
\paper Conformal invariants of hyperbolic planar domains
\jour Ufimsk. Mat. Zh.
\yr 2019
\vol 11
\issue 2
\pages 3--18
\jour Ufa Math. J.
\yr 2019
\vol 11
\issue 2
\pages 3--18

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    This publication is cited in the following articles:
    1. R. G. Nasibullin, R. V. Makarov, “Neravenstva Khardi s dopolnitelnymi slagaemymi i uravneniya tipa Lemba”, Sib. matem. zhurn., 61:6 (2020), 1377–1397  mathnet  crossref
    2. V R. Makarov , R. G. Nasibullin, “Hardy type inequalities and parametric Lamb equation”, Indag. Math.-New Ser., 31:4 (2020), 632–649  crossref  mathscinet  zmath  isi  scopus
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