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Subclasses of Biholomorphic Mappings Under the Extension Operators

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Abstract

In this article, we mainly study the invariance of some biholomorphic mappings with special geometric characteristics under the extension operators. First we generalize the Roper-Suffridge extension operators on Bergman-Hartogs domains. Then, by the geometric characteristics of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suffridge operators preserve the properties of SΩ* (β,A,B), parabolic and spirallike mappings of type β and order ρ, strong and almost spirallike mappings of type β and order α as well as almost starlike mappings of complex order λ on \(\Omega_{p_{1}}^{B^{n}},\ldots,_{p_{s},q}\) under different conditions, respectively. The conclusions provide new approaches to construct these biholomorphic mappings in several complex variables.

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References

  1. Roper K A, Suffridge T J. Convex mappings on the unit ball of Cn. J Anal Math, 1995, 65: 333–347

    Article  MathSciNet  MATH  Google Scholar 

  2. Graham I, Kohr G. Univalent mappings associated with the Roper-Suffridge extension operator. J Anal Math, 2000, 81: 331–342

    Article  MathSciNet  MATH  Google Scholar 

  3. Gong S, Liu T S. The generalized Roper-Suffridge extension operator. J Math Anal Appl, 2003, 284: 425–434

    Article  MathSciNet  MATH  Google Scholar 

  4. Muir J R. A modification of the Roper-Suffridge extension operator. Comput Methods Funct Theory, 2005, 5(1): 237–251

    Article  MathSciNet  MATH  Google Scholar 

  5. Elin M, Levenshtein M. Covering Results and Perturbed Roper-Suffridge Operators. Complex Anal Oper Theory, 2014, 8: 25–36

    Article  MathSciNet  MATH  Google Scholar 

  6. Kohr G. Loewner chains and a modification of the Roper-Suffridge extension operator. Mathematica, 2006, 71(1): 41–48

    MathSciNet  MATH  Google Scholar 

  7. Muir J R. A class of Loewner chain preserving extension operators. J Math Anal Appl, 2008, 337(2): 862–879

    Article  MathSciNet  MATH  Google Scholar 

  8. Elin M. Extension operators via semigroups. J Math Anal Appl, 2011, 377: 239–250

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu M S, Zhu Y C. The Extension Operator in Banach Spaces for Locally Biholomorphic Mappings. Acta Math Sci, 2008, 28B(3): 711–720

    MathSciNet  MATH  Google Scholar 

  10. Tang Y Y. Roper-Suffridge Operators on Bergman-Hartogs Domain. Kaifeng: Henan University, 2016

    Google Scholar 

  11. Pan L, Wang A. The holomorphic automorphism group of the domains of the Bergman-Hartogs type. Science China: Mathematics, 2015, 45(1): 31–42

    Google Scholar 

  12. Feng S X, Liu T S. The generalized Roper-Suffridge extension operator. Acta Math Sci, 2008, 28B: 63–80

    MathSciNet  MATH  Google Scholar 

  13. Liu X S, Feng S X. A remark on the generalized Roper-Suffridge extension operator for spirallike mappings of type β and order α. Chin Quart J of Math, 2009, 24(2): 310–316

    MathSciNet  MATH  Google Scholar 

  14. Liu X S, Liu T S. The generalized Roper-Suffridge extension operator on a Reinhardt domain and the unit ball in a complex Hilbert space. Chin Ann Math, 2005, 26A(5): 721–730

    MathSciNet  MATH  Google Scholar 

  15. Gao C L. The Generalized Roper-Suffridge Extension Operator on a Reinhardt Domain[D]. Jinhua: Zhejiang Normal University, 2012

    Google Scholar 

  16. Feng S X, Liu T S, Ren G B. The growth and covering theorems for several mappings on the unit ball in complex Banach spaces. Chin Ann Math, 2007, 28A(2): 215–230

    MathSciNet  MATH  Google Scholar 

  17. Zhu Y C, Liu M S. The generalized Roper-Suffridge extension operator on Reinhardt domain D p. Taiwanese J Math, 2010, 14(2): 359–372

    Article  MathSciNet  MATH  Google Scholar 

  18. Feng S X, Zhang X F, Chen H Y. Parabolic starlike mapping in several complex variables. Acta Math Sin (Chinese Series), 2011, 54(3): 467–482

    MathSciNet  MATH  Google Scholar 

  19. Cai R H, Liu X S. The third and fourth coefficient estimations for the subclasses of strongly spirallike functions. Journal of Zhanjiang Normal College, 2010, 31: 38–43

    Google Scholar 

  20. Zhao Yanhong. Almost Starlike Mappings of Complex Order λ on the Unit Ball of a Complex Banach Space[D]. Jinhua: Zhejiang Normal University, 2013

    Google Scholar 

  21. Liu T S, Ren G B. Growth theorem of convex mappings on bounded convex circular domains. Science in China, 1998, 41(2): 123–130

    Article  MathSciNet  MATH  Google Scholar 

  22. Ahlfors L V. Complex Analysis. 3rd ed. New York: Mc Graw-Hill Book Co, 1979

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. Cui Y, Wang C. Property of the generalized Roper-Suffridge extension operator on specific domains. Journal of Sichuan Normal University (Natural Science), 2013, 36(5): 726–729

    MathSciNet  MATH  Google Scholar 

  25. Cui Y, Wang C, Liu H. The generalized Roper-Suffrige operator on the unit ball in complex Banach and Hilbert spaces. Acta Math Sci, 2017, 37B(6): 1817–1829

    Article  MATH  Google Scholar 

  26. Wang J F. Modified Roper-Suffridge operator for some subclasses of starlike mappings on Reinhardt domains. Acta Math Sci, 2013, 33B(6): 1627–1638

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, 466001, China

    Chaojun Wang  (王朝军)

  2. College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, 466001, China

    Yanyan Cui  (崔艳艳)

  3. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050016, China

    Yanyan Cui  (崔艳艳)

  4. Institute of Contemporary Mathematics, Henan University, Kaifeng, 475001, China

    Hao Liu  (刘洁)

Authors
  1. Chaojun Wang  (王朝军)
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  2. Yanyan Cui  (崔艳艳)
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  3. Hao Liu  (刘洁)
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Corresponding author

Correspondence to Hao Liu  (刘洁).

Additional information

This work was supported by NSF of China (11271359, 11471098), Science and Technology Research Projects of Henan Provincial Education Department (19B110016), Scientific Research Innovation Fund Project of Zhoukou Normal University (ZKNUA201805).

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Wang, C., Cui, Y. & Liu, H. Subclasses of Biholomorphic Mappings Under the Extension Operators. Acta Math Sci 39, 297–311 (2019). https://doi.org/10.1007/s10473-019-0122-9

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  • DOI: https://doi.org/10.1007/s10473-019-0122-9

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