Advertisement

Acta Mathematica Scientia

, Volume 39, Issue 1, pp 297–311 | Cite as

Subclasses of Biholomorphic Mappings Under the Extension Operators

  • Chaojun Wang (王朝军)
  • Yanyan Cui (崔艳艳)
  • Hao Liu (刘洁)
Article

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.

Key words

spirallike mappings Roper-Suffridge operator Bergman-Hartogs domains 

2010 MR Subject Classification

32A30 30C25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Roper K A, Suffridge T J. Convex mappings on the unit ball of Cn. J Anal Math, 1995, 65: 333–347MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Graham I, Kohr G. Univalent mappings associated with the Roper-Suffridge extension operator. J Anal Math, 2000, 81: 331–342MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Gong S, Liu T S. The generalized Roper-Suffridge extension operator. J Math Anal Appl, 2003, 284: 425–434MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Muir J R. A modification of the Roper-Suffridge extension operator. Comput Methods Funct Theory, 2005, 5(1): 237–251MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Elin M, Levenshtein M. Covering Results and Perturbed Roper-Suffridge Operators. Complex Anal Oper Theory, 2014, 8: 25–36MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Kohr G. Loewner chains and a modification of the Roper-Suffridge extension operator. Mathematica, 2006, 71(1): 41–48MathSciNetzbMATHGoogle Scholar
  7. [7]
    Muir J R. A class of Loewner chain preserving extension operators. J Math Anal Appl, 2008, 337(2): 862–879MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Elin M. Extension operators via semigroups. J Math Anal Appl, 2011, 377: 239–250MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Liu M S, Zhu Y C. The Extension Operator in Banach Spaces for Locally Biholomorphic Mappings. Acta Math Sci, 2008, 28B(3): 711–720MathSciNetzbMATHGoogle Scholar
  10. [10]
    Tang Y Y. Roper-Suffridge Operators on Bergman-Hartogs Domain. Kaifeng: Henan University, 2016Google Scholar
  11. [11]
    Pan L, Wang A. The holomorphic automorphism group of the domains of the Bergman-Hartogs type. Science China: Mathematics, 2015, 45(1): 31–42Google Scholar
  12. [12]
    Feng S X, Liu T S. The generalized Roper-Suffridge extension operator. Acta Math Sci, 2008, 28B: 63–80MathSciNetzbMATHGoogle Scholar
  13. [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): 310–316MathSciNetzbMATHGoogle Scholar
  14. [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): 721–730MathSciNetzbMATHGoogle Scholar
  15. [15]
    Gao C L. The Generalized Roper-Suffridge Extension Operator on a Reinhardt Domain[D]. Jinhua: Zhejiang Normal University, 2012Google Scholar
  16. [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): 215–230MathSciNetzbMATHGoogle Scholar
  17. [17]
    Zhu Y C, Liu M S. The generalized Roper-Suffridge extension operator on Reinhardt domain D p. Taiwanese J Math, 2010, 14(2): 359–372MathSciNetCrossRefzbMATHGoogle Scholar
  18. [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): 467–482MathSciNetzbMATHGoogle Scholar
  19. [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: 38–43Google Scholar
  20. [20]
    Zhao Yanhong. Almost Starlike Mappings of Complex Order λ on the Unit Ball of a Complex Banach Space[D]. Jinhua: Zhejiang Normal University, 2013Google Scholar
  21. [21]
    Liu T S, Ren G B. Growth theorem of convex mappings on bounded convex circular domains. Science in China, 1998, 41(2): 123–130MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Ahlfors L V. Complex Analysis. 3rd ed. New York: Mc Graw-Hill Book Co, 1979zbMATHGoogle Scholar
  23. [23]
    Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York: Marcel Dekker, 2003zbMATHGoogle Scholar
  24. [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): 726–729MathSciNetzbMATHGoogle Scholar
  25. [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): 1817–1829CrossRefzbMATHGoogle Scholar
  26. [26]
    Wang J F. Modified Roper-Suffridge operator for some subclasses of starlike mappings on Reinhardt domains. Acta Math Sci, 2013, 33B(6): 1627–1638MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Wuhan Institutes of Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  • Chaojun Wang (王朝军)
    • 1
  • Yanyan Cui (崔艳艳)
    • 2
    • 3
  • Hao Liu (刘洁)
    • 4
  1. 1.College of Mathematics and StatisticsZhoukou Normal UniversityZhoukouChina
  2. 2.College of Mathematics and StatisticsZhoukou Normal UniversityZhoukouChina
  3. 3.College of Mathematics and Information ScienceHebei Normal UniversityShijiazhuangChina
  4. 4.Institute of Contemporary MathematicsHenan UniversityKaifengChina

Personalised recommendations