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A Note on Pluriharmonic Functions in the Unit Polydisc in ℂn

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Abstract

We prove Schwarz-Pick lemma for bounded and strictly positive pluriharmonic functions in the unit polydisc in ℂn. We give a distance estimate in terms of Kobayashi metric as well as estimates on the gradient and \(\mathcal {M}\)-invariant real gradient of such functions. All of our estimates are sharp.

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References

  1. M. Arsenović and J. Gajić, A note on positive pluriharmonic functions in the unit ball in Cn, Bull. Int. Math. Virtual Inst., 11 (2021), 2021–249.

    MATH  Google Scholar 

  2. S. Chen and A. Rasila, Schwarz-Pick type estimates of pluriharmonic map**s in the unit polydisk, Illinois J. Math., 58 (2014), 2014–1015.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, 2nd extended ed., De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter (2013)

  4. A. Khalfallah, Old and new invariant pseudo-distances defined by pluriharmonic functions, Complex Anal. Oper. Theory, 9 (2015), 2015–113.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Knese, A Schwarz lemma on the polydisk, Proc. Amer. Math. Soc., 135 (2007), 2007–2759.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Map**s. An Introduction, World Scientific Publishing Co. (Hackensack, NJ, 2005).

    Book  MATH  Google Scholar 

  7. S. Krantz, Function Theory of Several Complex Variables, 2nd ed., American Mathematical Society (Providence, RI, 2001).

    MATH  Google Scholar 

  8. M. Mateljević, Schwarz lemma and Kobayashi metrics for harmonic and holomorphic functions, J. Math. Anal. Appl., 464 (2018), 2018–78.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Mateljević and M. Svetlik, Hyperbolic metric on the strip and the Schwarz lemma for HQR map**s, Appl. Anal. Discrete Math., 14 (2020), 2020–150.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Melentijević, Invariant gradient in refinements of Schwarz lemma and Harnack inequalities, Ann. Acad. Sci. Fenn. Math., 43 (2018), 2018–391.

    Article  MathSciNet  MATH  Google Scholar 

  11. W. Rudin, Function Theory in Polydiscs, W. A. Benjamin, Inc. (New York-Amsterdam, 1969).

    MATH  Google Scholar 

  12. Z. Xu, Schwarz lemma for pluriharmonic functions, Indag. Math. (N.S.), 27 (2016), 2016–923.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

The author would like to thank Professor Miloš Arsenović for his valuable suggestions which have greatly improved the presentation of the article.

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Authors and Affiliations

  1. Faculty Of Natural Sciences and Mathematics, University of Banja Luka, 78000, Banja Luka, Republic of Srpska, Bosnia and Herzegovina

    J. Gajić

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  1. J. Gajić
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Correspondence to J. Gajić.

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Gajić, J. A Note on Pluriharmonic Functions in the Unit Polydisc in ℂn. Anal Math 48, 1047–1054 (2022). https://doi.org/10.1007/s10476-022-0166-2

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  • DOI: https://doi.org/10.1007/s10476-022-0166-2

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