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Equivalent moduli of continuity, Bloch’s theorem for pluriharmonic mappings in \(\mathbb{B}{\kern-8pt}\mathbb{B}^{\boldsymbol n}\)

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Abstract

In this paper, we first establish a Schwarz–Pick type theorem for pluriharmonic mappings and then we apply it to discuss the equivalent norms on Lipschitz-type spaces. Finally, we obtain several Landau’s and Bloch’s type theorems for pluriharmonic mappings.

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References

  1. Ahlfors L V, An extension of Schwarz’s lemma, Trans. Am. Math. Soc. 43 (1938) 359–364

    MathSciNet  Google Scholar 

  2. Arsenović M, Boz̆in V and Manojlović V, Moduli of continuity of harmonic quasiregular mappings in \(\mathbb{B}^{n}\), Potential Anal. 34(3) (2011) 283–291

    Article  MathSciNet  MATH  Google Scholar 

  3. Bochner S, Bloch’s theorem for real variables, Bull. Am. Math. Soc. 52(1946) 715–719

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen H, The Schwarz–Pick lemma for planar harmonic mappings, Sci. China Math. 54 (2011) doi:10.1007/s11425-011-4193-x

    Google Scholar 

  5. Chen H and Gauthier P M, Bloch constants in several variables, Trans. Am. Math. Soc. 353 (2001) 1371–1386

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen H and Gauthier P M, The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings, Proc. Am. Math. Soc. 139 (2011), 583–595

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen Sh, Ponnusamy S and Wang X, On planar harmonic Lipschitz and planar harmonic hardy classes, Ann. Acad. Sci. Fenn. Math. 36 (2011) 567–576

    Article  MathSciNet  MATH  Google Scholar 

  8. Dyakonov K M, Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math. 178 (1997) 143–167

    Article  MathSciNet  MATH  Google Scholar 

  9. Dyakonov K M, Holomorphic functions and quasiconformal mappings with smooth moduli, Adv. Math. 187 (2004) 146–172

    Article  MathSciNet  MATH  Google Scholar 

  10. Fitzgerald C H and Thomas C R, Some bounds on convex mappings in several complex variables, Pacific J. Math. 165 (1994) 295–320

    Article  MathSciNet  MATH  Google Scholar 

  11. Gehring F W and Martio O, Lipschitz-classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. 10 (1985) 203–219

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Kaufman R and Wu J M, Distances and the Hardy-Littlewood property, Complex Variables Theory Appl. 4 (1984) 1–5

    Article  MathSciNet  MATH  Google Scholar 

  14. Knez̆ević M and Mateljević M, On the quasi-isometries of harmonic quasiconformal mappings, J. Math. Anal. Appl. 334 (2007) 404–413

    Article  MathSciNet  Google Scholar 

  15. Kojić V and Pavlović M, Subharmonicity of ∣ f ∣ p for quasiregular harmonic functions with applications, J. Math. Anal. Appl. 342 (2008) 742–746

    Article  MathSciNet  MATH  Google Scholar 

  16. Lappalainen V, Lip h -extension domains, Ann. Acad. Sci. Fenn. 56 (1985)

  17. Liu X Y, Bloch functions of several complex variables, Pacific J. Math. 152 (1992) 347–363

    Article  MathSciNet  MATH  Google Scholar 

  18. Mateljević M, A version of Bloch theorem for quasiregular harmonic mappings, Rev. Roum. Math. Pures. Appl. 47 (2002) 705–707

    MATH  Google Scholar 

  19. Mateljević M, Versions of Koebe 1/4-theorem for analytic and quasiregular harmonic functions and applications, Publ. Inst. Math. (Beograd) (N.S.) 84 (2008) 61–72

    Article  MathSciNet  Google Scholar 

  20. Mateljević M, Quasiconformal and quasiregular harmonic analogues of Koebe’s theorem and applications, Ann. Acad. Sci. Fenn. Math. 32 (2007) 301–315

    MathSciNet  MATH  Google Scholar 

  21. Mateljević M and Vuorinen M, On harmonic quasiconformal quasi-isometries, J. Inequal. Appl., vol. 2010, Article ID 178732, 19 pages, doi:10.1155/2010/1787 (2010)

  22. Pavlović M, On Dyakonov’s paper equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math. 183 (1999) 141–143

    Article  MathSciNet  MATH  Google Scholar 

  23. Rudin W, Function theory in the unit ball of ℂn (1980) (New York, Heidelberg, Berlin: Spring-Verlag)

    Google Scholar 

  24. Takahashi S, Univalent mappings in several complex variables, Ann. Math. 53 (1951) 464–471

    Article  MATH  Google Scholar 

  25. Vuorinen M, Conformal geometry and quasiregular mappings, Lecture Notes in Math., vol. 1319 (1988) (Springer-Verlag)

  26. Wu H, Normal families of holomorphic mappings, Acta Math. 119 (1967) 193–233

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was partly supported by NSF of China (No. 11071063). The work was carried out while the first author was visiting IIT Madras, under ‘RTFDCS Fellowship’. He also would like to thank Centre for International Co-operation in Science (formerly Centre for Co-operation in Science & Technology among Developing Societies (CCSTDS)) for its support and co-operation.

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  1. SH CHEN

    Present address: Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, Hunan, 421008, People’s Republic of China

Authors and Affiliations

  1. Department of Mathematics, Hunan Normal University, Changsha, Hunan, 410081, People’s Republic of China

    SH CHEN & X WANG

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    S PONNUSAMY

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  1. SH CHEN
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Correspondence to X WANG.

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CHEN, S., PONNUSAMY, S. & WANG, X. Equivalent moduli of continuity, Bloch’s theorem for pluriharmonic mappings in \(\mathbb{B}{\kern-8pt}\mathbb{B}^{\boldsymbol n}\) . Proc Math Sci 122, 583–595 (2012). https://doi.org/10.1007/s12044-012-0095-2

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