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Schwarz–Pick Lemma for Harmonic and Hyperbolic Harmonic Functions

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Abstract

We establish some inequalities of Schwarz–Pick type for harmonic and hyperbolic harmonic functions on the unit ball of \({\mathbb {R}^n}\) and we disprove a recent conjecture of Liu (Int Math Res Not, 2021).

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References

  1. Liu, C.: A proof of the Khavinson conjecture. Math. Ann. 380, 719–732 (2021)

    Article  MathSciNet  Google Scholar 

  2. Khavinson, D.: An extremal problem for harmonic functions in the ball. Canadian Math. Bull. 35, 218–220 (1992)

    Article  MathSciNet  Google Scholar 

  3. Kresin, G., Maz’ya, V.: Sharp pointwise estimates for directional derivatives of harmonic function in a multidimensional ball. J. Math. Sci. 169, 167–187 (2010)

    Article  MathSciNet  Google Scholar 

  4. Kresin, G., Maz’ya, V.: Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete Contin. Dyn. Syst. 28, 425–440 (2010)

    Article  MathSciNet  Google Scholar 

  5. Kresin, G., Maz’ya, V.: Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems. Mathematical Surveys and Monographs, vol. 183. American Mathematical Society, Providence, RI (2012)

    MATH  Google Scholar 

  6. Marković, M.: Solution to the Khavinson problem near the boundary of the unit ball. Constr. Approx. 45(2), 243–271 (2017)

    Article  MathSciNet  Google Scholar 

  7. Mateljević, M., Khalfallah, A.: On some Schwarz type inequalities. J. Inequal. Appl. 2020, 164 (2020)

    Article  MathSciNet  Google Scholar 

  8. Melentijević, P.: A proof of the Khavinson conjecture in \(\mathbb{R}^3\). Adv. Math. 352, 1044–1065 (2019)

    Article  MathSciNet  Google Scholar 

  9. Liu, C.: Schwarz-Pick Lemma for harmonic functions. Int. Math. Res. Not. (2021). https://doi.org/10.1093/imrn/rnab158

    Article  Google Scholar 

  10. Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Graduate Texts in Mathematics, vol. 137. Springer, New York (1992)

    Book  Google Scholar 

  11. Burgeth, B.: A Schwarz lemma for harmonic and hyperbolic harmonic functions in higher dimensions. Manuscr. Math. 77(2–3), 283–291 (1992)

    Article  MathSciNet  Google Scholar 

  12. Colonna, F.: The Bloch constant of bounded harmonic mappings. Indiana Univ. Math. J. 38, 829–840 (1989)

    Article  MathSciNet  Google Scholar 

  13. Kalaj, D., Vuorinen, M.: On harmonic functions and the Schwarz lemma. Proc. Am. Math. Soc. 140(1), 161–165 (2012)

    Article  MathSciNet  Google Scholar 

  14. Burgeth, B.: Schwarz lemma type inequalities for harmonic functions in the ball. Class. Mod. Potent. Theory Appl. (x):133–147 (1994)

  15. Khalfallah, A., Mateljević, M., Mhamdi, M.: Some properties of mappings admitting general Poisson representations. Med. J. Math. 18, 1–19 (2021)

    MathSciNet  MATH  Google Scholar 

  16. Borgwardt, K.H.: The Simplex Method. Springer, Berlin (1987)

    Book  Google Scholar 

  17. Alzer, H.: Inequalities for the volume of the unit ball in \(\mathbb{R}^n\). J. Math. Anal. Appl. 252, 353–363 (2000)

    Article  MathSciNet  Google Scholar 

  18. Khalfallah, A., Haggui, F., Mateljević, M.: Khavinson conjecture for hyperbolic harmonic functions on the unit ball. J. Math. Anal. Appl. 513(2), 126241 (2022)

  19. Pavlović, M.: A Schwarz lemma for the modulus of a vector-valued analytic function. Proc. Amer. Math. Soc. 139, 969–973 (2011)

    Article  MathSciNet  Google Scholar 

  20. Kalaj, D.: Heinz–Schwarz inequalities for harmonic mappings in the unit ball. Ann. Acad. Sci. Fenn. Math. 41, 457–464 (2016)

    Article  MathSciNet  Google Scholar 

  21. Chen, H.: The Schwarz–Pick lemma and Julia lemma for real planar harmonic mappings. Sci. China Math. 56, 2327–2334 (2013)

    Article  MathSciNet  Google Scholar 

  22. Mateljević, M.: Schwarz lemma and Kobayashi metrics for harmonic and holomorphic functions. J. Math. Anal. Appl. 464(1), 78–100 (2018)

    Article  MathSciNet  Google Scholar 

  23. Chen, Sh., Hamada, H.: Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions. J. Funct. Anal 282(1), 109254 (2022)

    Article  MathSciNet  Google Scholar 

  24. Dai, S.Y., Chen, H.H., Pan, Y.F.: The high order Schwarz–Pick lemma on complex Hilbert balls. Sci. China Math. (2010). https://doi.org/10.1007/s11425-010-0119-2

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhu, J.-F.: Schwarz–Pick Type estimates for gradients of pluriharmonic mappings of the unit ball. Results Math. 74, 1–16 (2019)

    Article  MathSciNet  Google Scholar 

  26. Ahlfors, L.: Möbius transformation in several dimensions. University of Minnesota, School of mathematics (1981)

  27. Alzer, H.: Inequalities for the volume of the unit ball in \(\mathbb{R}^n\), II. Med. J. M. 5, 395–413 (2008)

    MATH  Google Scholar 

  28. Mortici, C.: Monotonicity properties of the volume of the unit ball in \(\mathbb{R}^n\). Optim. Lett. 4, 457–464 (2010)

    Article  MathSciNet  Google Scholar 

  29. Stoll, M.: Harmonic and Subharmonic Function Theory on the Hyperbolic Ball. London Mathematical Society Lecture Note Series, vol. 431. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Adel Khalfallah

  2. Faculty of Mathematics, University of Belgrade, Studentski Trg 16, Belgrade, Republic of Serbia

    Miodrag Mateljević & Bojana Purtić

Authors
  1. Adel Khalfallah
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  2. Miodrag Mateljević
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  3. Bojana Purtić
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Correspondence to Adel Khalfallah.

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Khalfallah, A., Mateljević, M. & Purtić, B. Schwarz–Pick Lemma for Harmonic and Hyperbolic Harmonic Functions. Results Math 77, 167 (2022). https://doi.org/10.1007/s00025-022-01686-5

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