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Schwarz Lemma, and Distortion for Harmonic Functions Via Length and Area

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Abstract

We give sharp estimates for distortion of harmonic mappings u from the unit disc \(\mathbb {U}\) into \(\mathbb {R}^{m}\), at a prescribed point by means of diameter and area of the corresponding surface \(S=u(\mathbb {U})\), and via the generalized length of the boundary of S.

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Notes

  1. Using sticky-notes he wrote numerous comments on margins of the manuscript in which the corrections are elaborated.

References

  1. Axler, S., Bourdon, P., Ramey, W.: Harmonic function theory. Springer-Verlag, New York (1992)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Duren, P: Theory of Hp-spaces. Academic Press, Cambridge (1970)

    Google Scholar 

  4. Kalaj, D.: A sharp inequality for harmonic diffeomorphisms of the unit disk, arXiv:1706.01990v1 [math.CV] 6 Jun 2017, The Journal of Geometric Analysis, pp 110, First Online: 14. https://link.springer.com/article/10.1007/s12220-018-9996-3 (2018)

  5. Kalaj, D.: Minimal surfaces and Schwarz lemma. arXiv:1708.01848v1 [math.CV] 6 (2017)

  6. Kalaj, D.: A sharp inequality for diffeomorphisms of the unit disk, Communication at VIII SimpozijumMatematika i primene 2017, 17 i 18 nov 2017, Beograd, Abstracts. http://alas.matf.bg.ac.rs/konferencija/s2017/kalabs.pdf

  7. Kalaj, D., Marković, M., Mateljević, M.: Caratheodory and Smirnov type theorems for harmonic mappings of the unit disk onto surfaceś. Ann. Acad. Sci. Fenn. Math. 38(2), 565–580 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Krantz, S. G.: The Carathéodory and Kobayashi metrics and applications in complex analysis. arXiv:0608772v1 [math.CV] 31 (2006)

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

    Article  MathSciNet  Google Scholar 

  11. Kresin, G., Maz’ya, V.: Maximum principles and sharp constants for solutions of elliptic and parabolic systems (2014)

  12. Mateljević, M.: (a) Schwarz lemma and Kobayashi Metrics for holomorphic and pluriharmonic functions, arXiv:1704.06720v1 [math.CV] 21 Apr 2017 (b) Schwarz lemma, Kobayashi Metrics and FPT preprintNovember (2016)

  13. Mateljević, M.: Communications at analysis seminar, University of Belgrade, Belgrade

  14. Mateljević, M.: Schwarz lemma and Kobayashi Metrics for harmonicand holomorphic functions, J. Math. Anal. Appl. https://doi.org/10.1016/j.jmaa.2018.03.069(2018)

  15. (a) https://www.researchgate.net/project/Schwarz-lemma-the-Caratheodory-and-Kobayashi-Metrics-and-Applications-in-Complex-Analysishttps://www.researchgate.net/project/Schwarz-lemma-the-Caratheodory-and-Kobayashi-Metrics-and-Applications-in-Complex-Analysis, Miodrag Mateljevic, Aug 10, 2016 (b) https://www.researchgate.net/post/What_are_the_most_recent_versions_of_The_Schwarz_Lemma [accessed Jul 31, 2017]. How to solve an exremal problems related to harmonic functions?. Available from: https://www.researchgate.net/post/How_to_solve_a_exremal_problems_related_to_harmonic_functions [accessed Aug 3, 2017]. (c) Rigidity of holomorphic mappings & Schwarz and Jack lemma, https://doi.org/10.13140/RG.2.2.34140.90249, https://www.researchgate.net/publication/325430073_Miodrag_Mateljevic_Rigidity_of_holomorphic_mappings_Schwarz_and_Jack_lemma

  16. Mateljević, M.: Schwarz lemma and distortion for harmonic functions via length and area arXiv:1805.02979v1 [math.CV] 8 (2018)

  17. Mateljević, M., Khalfallah, A.: Schwarz lemmas for mappings with bounded Laplacian, arXiv:1810.08823v1 [math.CV]

  18. Mateljević, M.: Schwarz type inequalities for harmonic and related functions in the disk and the ball. In: Lecko, A. (ed.) Current Research in Mathematicaland Computer Sciences IIPublisher UWM, Olsztyn, pp. 157–194 (2018)

  19. Partyka, D., Sakan, K.: Quasiconformal and Lipschitz harmonic mappings of the unit disc onto bounded convex domains. Ann. Acad. Sci. Fenn., Math. 39, 811–830 (2014)

    Article  MathSciNet  Google Scholar 

  20. Chen, H.: The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings. Sci. China Math. 56(11), 2327–2334 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We wish to thank Shi Qingtian, who has been reading very carefully several versions of this manuscript, for useful discussions and useful comments. In particular, we are really grateful to an anonymous reviewer, who has put a lot of effort into examining closely the text,Footnote 1 for providing insightful comments and directions for improvement of the exposition.

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  1. Faculty of mathematics, University of Belgrade, Studentski Trg 16, Belgrade, Serbia

    M. Mateljević

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  1. M. Mateljević
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Correspondence to M. Mateljević.

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Mateljević, M. Schwarz Lemma, and Distortion for Harmonic Functions Via Length and Area. Potential Anal 53, 1165–1190 (2020). https://doi.org/10.1007/s11118-019-09802-x

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