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Schwarz Lemma at the Boundary of the Unit Ball in \(\mathbb {C}^n\) and Its Applications

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Abstract

In this paper, we first establish a new type of the classical boundary Schwarz lemma for holomorphic self-mappings of the unit ball in \(\mathbb {C}^n\). We then apply our new Schwarz lemma to study problems from the geometric function theory in several complex variables.

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References

  1. Abouhajar, A., White, M., Young, N.: A Schwarz lemma for a domain related to \(\mu \)-synthesis. J. Geom. Anal. 17(4), 717–750 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Yau, S.: A general Schwarz lemma for Kähler manifolds. Amer. J. Math. 100(1), 197–203 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  3. Siu, Y., Yeung, S.: Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees. Am. J. Math. 119(5), 1139–1172 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Graham, I., Minda, D.: A Schwarz lemma for multivalued functions and distortion theorems for Bloch functions with branch point. Trans. Am. Math. Soc. 351(12), 4741–4752 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Tsuji, H.: A general Schwarz lemma. Math. Ann. 256(3), 387–390 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kim, K., Lee, H.: Schwarz’s lemma from a differential geometric viewpoint. IISc Press, Bangalore (2011)

    MATH  Google Scholar 

  7. Rodin, B.: Schwarz’s lemma for circle packings. Invent. Math. 89(2), 271–289 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ahlfors, L.: An extension of Schwarz’s lemma. Trans. Am. Math. Soc. 43(3), 359–364 (1938)

    MathSciNet  Google Scholar 

  9. Garnett, J.: Bounded Analytic Functions. Academic Press, New York (1981)

    MATH  Google Scholar 

  10. Bonk, M.: On Bloch’s constant. Proc. Am. Math. Soc. 110(4), 889–894 (1990)

    MATH  MathSciNet  Google Scholar 

  11. Liu, T., Ren, G.: The growth theorem of convex mappings on bounded convex circular domains. Sci. China Ser. A 41(2), 123–130 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gong, S., Liu, T.: Distortion theorems for biholomorphic convex mappings on bounded convex circular domains. Chin. Ann. of Math. 20B(3), 297–304 (1999)

    Article  MathSciNet  Google Scholar 

  13. Zhang, W., Liu, T.: On growth and covering theorems of quasi-convex mappings in the unit ball of a complex Banach space. Sci. China Ser. A 45(12), 1538–1547 (2002)

    MATH  MathSciNet  Google Scholar 

  14. Krantz, S.: The Schwarz lemma at the boundary. Complex Var. Elliptic Equ. 56(5), 455–468 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chelst, D.: A generalized Schwarz lemma at the boundary. Proc. Am. Math. Soc. 129(11), 3275–3278 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Osserman, R.: A sharp Schwarz inequality on the boundary. Proc. Am. Math. Soc. 128, 3513–3517 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Burns, D., Krantz, S.: Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J. Am. Math. Soc. 7, 661–676 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Huang, X.: A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains. Can. J. Math. 47(2), 405–420 (1995)

    Article  MATH  Google Scholar 

  20. Huang, X.: On a semi-rigidity property for holomorphic maps. Asian J. Math. 7(4), 463–492 (2003)

    MATH  MathSciNet  Google Scholar 

  21. Gong, S.: Convex and Starlike Mappings in Several Complex Variables. Springer Press/Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  22. Krantz, S.: Function Theory of Several Complex Variables. Wiley, New York (1982)

    MATH  Google Scholar 

  23. Cartan, H.: Sur la possibilité d’éntendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes. In: Montel, P., (ed) Leçons sur les Fonctions Univalentes ou Multivalentes, pp. 129–155. Gauthier-Villar, Cambridge, MA (1933)

  24. Barnard, R., FitzGerald, C., Gong, S.: A distortion theorem for biholomorphic mappings in \({\mathbb{C}}^2\). Trans. Am. Math. Soc. 344, 907–924 (1994)

    MATH  MathSciNet  Google Scholar 

  25. Liu, T., Zhang, W.: A distortion theorem for biholomorphic convex mappings in \({\mathbb{C}}^n\). Chin. J. Contemp. Math. 20(3), 351–360 (1999)

    Google Scholar 

  26. Barnard, R., FitzGerald, C., Gong, S.: The growth and \(\frac{1}{4}\)-theorems for starlike mappings in \({\mathbb{C}}^n\). Pac. J. Math. 150, 13–22 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  27. Liu, T., Ren, G.: The growth theorem for starlike mappings on bounded starlike circular domains. Chin. Ann. Math. Ser. 19B(4), 401–408 (1998)

    MathSciNet  Google Scholar 

  28. Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Marcel Dekker, New York (2003)

    MATH  Google Scholar 

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Acknowledgments

This work is supported by the NNSF of China (Nos. 11031008, 11101139, 11271124, 11001246), NSF of Zhejiang province (Nos.Y14A010047, Y6110260).

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Authors and Affiliations

  1. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, Zhejiang, People’s Republic of China

    Taishun Liu & Xiaomin Tang

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, People’s Republic of China

    Jianfei Wang

Authors
  1. Taishun Liu
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  2. Jianfei Wang
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  3. Xiaomin Tang
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Correspondence to Taishun Liu.

Additional information

Communicated by Alexander Isaev.

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Liu, T., Wang, J. & Tang, X. Schwarz Lemma at the Boundary of the Unit Ball in \(\mathbb {C}^n\) and Its Applications. J Geom Anal 25, 1890–1914 (2015). https://doi.org/10.1007/s12220-014-9497-y

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