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A submultiplicative property of the Carathéodory metric on planar domains

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Abstract

Given a pair of smoothly bounded domains \(D_1, D_2 \subset \mathbb {C}\), the purpose of this paper is to obtain an inequality that relates the Carathéodory metrics on \(D_1, D_2, D_1 \cap D_2\) and \(D_1 \cup D_2\).

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References

  1. Burbea J, The curvatures of the analytic capacity, J. Math. Soc. Japan29(4) (1977) 755–761

    Article  MathSciNet  Google Scholar 

  2. Jarnicki M and Pflug P, Invariant distances and metrics in complex analysis, extended ed., De Gruyter Expositions in Mathematics, vol. 9 (2013) (KG Berlin: Walter de Gruyter GmbH & Co)

  3. Kraus D and Roth O, Strong submultiplicativity of the Poincaré metric, J. Anal.24(1) (2016) 39–50

    Article  MathSciNet  Google Scholar 

  4. Lars A and Beurling A, Conformal invariants and function-theoretic null-sets, Acta Math.83 (1950) 101–129

    Article  MathSciNet  Google Scholar 

  5. Sarkar A D and Verma K, Boundary behaviour of some conformal invariants on planar domains, preprint available at arXiv:1904.06867.pdf

  6. Suita N, On a metric induced by analytic capacity, Kōdai Math. Sem. Rep.25 (1973) 215–218

    Article  MathSciNet  Google Scholar 

  7. Suita N, On a metric induced by analytic capacity. II, Kōdai Math. Sem. Rep.27(1-2) (1976) 159–162

  8. Solynin A Yu, Ordering of sets, hyperbolic metric, and harmonic measure, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) vol. 237 (1997), Anal. Teor. Chisel i Teor. Funkts.14, 129–147, 230

  9. Solynin A Yu, Elliptic operators and Choquet capacities, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) vol. 371 (2009), Analiticheskaya Teoriya Chisel i Teoriya Funktsiĭ. 24, 149–156, 178–179

  10. Younsi M, On the analytic and Cauchy capacities, J. Anal. Math.135(1) (2018) 185–202

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, 560 012, India

    Amar Deep Sarkar & Kaushal Verma

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  1. Amar Deep Sarkar
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  2. Kaushal Verma
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Correspondence to Amar Deep Sarkar.

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Communicating Editor: Parameswaran Sankaran

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Sarkar, A.D., Verma, K. A submultiplicative property of the Carathéodory metric on planar domains. Proc Math Sci 130, 35 (2020). https://doi.org/10.1007/s12044-020-00565-9

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  • DOI: https://doi.org/10.1007/s12044-020-00565-9

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