Advertisement

Acta Mathematica Scientia

, Volume 39, Issue 4, pp 1103–1120 | Cite as

The Invariance of Subclasses of Biholomorphic Mappings on Bergman-Hartogs Domains

  • Yanyan Cui (崔艳艳)
  • Hao Liu (刘浩)Email author
Article
  • 15 Downloads

Abstract

We mainly discuss the invariance of some subclasses of biholomorphic mappings under the generalized Roper-Suffridge operators on Bergman-Hartogs domains which are based on the unit ball Bn. Using the geometric properties and the distortion results of subclasses of biholomorphic mappings, we obtain the geometric characters of almost spirallike mappings of type β and order \(\alpha, S_\Omega^*(\beta, A, B)\), strong and almost spirallike mappings of type β and order α maintained under the generalized Roper-Suffridge operators on Bergman-Hartogs domains. Sequentially, we conclude that the generalized operators and the known operators preserve the same properties under some conditions. The conclusions generalize some known results.

Key words

Biholomorphic mappings spirallike mappings Bergman-Hartogs domain Roper-Suffridge operator 

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 Analyse Math, 2000, 81: 331–342MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Liu T S, Gong S. Family of ε starlike mappings (I). Chin Ann Math, 2002, 23A(3): 273–282MathSciNetzbMATHGoogle Scholar
  4. [4]
    Liu X S, Liu T S. On the generalized Roper-Suffridge extension operator for spirallike mappings of type β and order α. Chin Ann Math, 2006, 27A(6): 789–798MathSciNetzbMATHGoogle Scholar
  5. [5]
    Feng S X, Zhang X F, Chen H Y. The generalized Roper-Suffridge extension operator. Chin Ann Math, 2011, 54A(3): 467–482Google Scholar
  6. [6]
    Wang J, Liu T. The Roper-Suffridge extension operator and its applications to convex mappings in ℂ2. Trans Amer Math Soc, 2018. DOI:  https://doi.org/10.1090/tran/7221
  7. [7]
    Wang J F, Liu T S. A modification of the Roper-Suffridge extension operator for some holomorphic mappings. Chin Ann Math, 2010, 31A(4): 487–496zbMATHGoogle Scholar
  8. [8]
    Muir J R. A modification of the Roper-Suffridge extension operator. Comput Methods Funct Theory, 2005, 5(1): 237–251MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Muir J R, Suffridge T J. Unbounded convex mappings of the ball in ℂn. Trans Amer Math Soc, 2001, 129: 3389–3393MathSciNetzbMATHGoogle Scholar
  10. [10]
    Kohr G. Loewner chains and a modification of the Roper-Suffridge extension operator. Mathematica, 2006, 71(1): 41–48MathSciNetzbMATHGoogle Scholar
  11. [11]
    Muir J R. A class of Loewner chain preserving extension operators. J Math Anal Appl, 2008, 337(2): 862–879MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Tang Y Y. Roper-Suffridge operators on Bergman-Hartogs domain[D]. Kaifeng: Henan University Master Thesis, 2016Google Scholar
  13. [13]
    Zhu Y C, Liu M S. The generalized Roper-Suffridge extension operator on Reinhardt domain D p. Taiwanese Jour of Math, 2010, 14(2): 359–372CrossRefzbMATHGoogle Scholar
  14. [14]
    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
  15. [15]
    Gao C L. The generalized Roper-Suffridge operators on Reinhardt domains[D]. Jinhua: Zhejiang Normal University Master Thesis, 2012Google Scholar
  16. [16]
    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
  17. [17]
    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(6): 38–43Google Scholar
  18. [18]
    Liu T S, Ren G B. Growth theorem for starlike mappings on bounded starlike circular domains. Chin Ann of Math, 1998, 19B(4): 401–408MathSciNetzbMATHGoogle Scholar
  19. [19]
    Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York: Marcel Dekker, 2003zbMATHGoogle Scholar
  20. [20]
    Zhang J, Lu J. Distortion theorems of almost spirallike mappings of type β with order α on the unit polydisc. Journal of Huzhou Teachers College, 2011, 33(2): 46–50zbMATHGoogle Scholar
  21. [21]
    Duren P L. Univalent Functions. New York: Springer-Verlag, 1983zbMATHGoogle Scholar
  22. [22]
    Ahlfors L V. Complex Analysis. 3rd ed. New York: Mc Graw-Hill Book Co, 1979zbMATHGoogle Scholar
  23. [23]
    Hamada H, Kohr G. The growth theorem and quasiconformal extension of strongly spirallike mappings of type α. Complex Variables Theory and Application, 2001, 44(4): 281–297MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Cui Y, Wang C, Liu H, et al. The distortion upper bounds for strongly spirallike mappings of type α. Mathematics in Practice and Theory, 2015, 45(13): 258–262MathSciNetzbMATHGoogle Scholar
  25. [25]
    Wang C, Cui Y, Liu H. Property of the Modified Roper-Suffridge Extension Operators on Bn. Acta Mathematica Sinica, 2016, 59(6): 729–744MathSciNetzbMATHGoogle Scholar
  26. [26]
    Wang C, Cui Y, Liu H. Properties of the modified Roper-Suffridge extension operators on Reinhardt domains. Acta Mathematica Scientia, 2016, 36B(6): 1767–1779MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Wang C, Liu A, Cui Y, et al. Subclasses of spirallike mappings on B n and the generalized Roper-Suffridge extension operators. Journal of Sichuan Normal University (Natural Science), 2016, 39(2): 231–235zbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsZhoukou Normal UniversityZhoukouChina
  2. 2.Institute of Contemporary MathematicsHenan UniversityKaifengChina

Personalised recommendations