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Results in Mathematics

, 74:60 | Cite as

On the Coefficient Inequality on a Bounded Starlike Circular Domain in \(\mathbb {C}^n\)

  • Qinghua XuEmail author
Article
  • 38 Downloads

Abstract

Let \(\Omega \) be a bounded starlike circular domain in \(\mathbb {C}^n\). In this paper, we introduce a class of holomorphic mappings \(\mathcal {M}_g\) on \(\Omega \). Let F(z) be a normalized locally biholomorphic mapping on \(\Omega \) such that \(J_F^{-1}(z)F(z)\in \mathcal {M}_g\). We obtain the Fekete and Szegö inequality for F(z). These results unify and generalize many known results.

Keywords

Coefficient inequality bounded starlike circular domain sharp coefficient bound 

Mathematics Subject Classification

Primary 32H02 Secondary 30C45 

Notes

References

  1. 1.
    Bracci, F.: Shearing process and an example of a bounded support function in \(S^0({\mathbb{B}}^2)\). Comput. Methods Funct. Theory 15, 151–157 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bracci, F., Graham, I., Hamada, H., Kohr, G.: Variation of Loewner chains, extreme and support points in the class \(S^0\) in higher dimensions. Constr. Approx. 43, 231–251 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fekete, M., Szegö, G.: Eine Bemerkunguber ungerade schlichte Funktionen. J. Lond. Math. Soc. 8, 85–89 (1933)CrossRefGoogle Scholar
  4. 4.
    Graham, I., Hamada, H., Kohr, G.: Parametric representation of univalent mappings in several complex variables. Canadian J. Math. 54, 324–351 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Graham, I., Kohr, G., Kohr, M.: Loewner chains and parametric representation in several complex variables. J. Math. Anal. Appl. 281, 425–438 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Graham, I., Hamada, H., Honda, T., Kohr, G., Shon, K.H.: Growth, distortion and coefficient bounds for Carathéodory families in \({\mathbb{C}}^n\) and complex Banach spaces. J. Math. Anal. Appl. 416, 449–469 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Graham, I., Hamada, H., Kohr, G., Kohr, M.: Support points and extreme points for mappings with \(A\)-parametric representation in \({\mathbb{C}}^n\). J. Geom. Anal. 26, 1560–1595 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Graham, I., Hamada, H., Kohr, G., Kohr, M.: Bounded support points for mappings with \(g\)- parametric representation in \({\mathbb{C}}^2\). J. Math. Anal. Appl. 454, 1085–1105 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guo, S.T., Xu, Q.H.: On the coefficient inequality for a subclass of starlike mappings in several complex variables. Chin. Q. J. Math. 33, 98–110 (2018)zbMATHGoogle Scholar
  10. 10.
    Hamada, H., Kohr, G.: Simple criterions for strongly starlikeness and starlikeness of certain order. Math. Nachr. 254(255), 165–171 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hamada, H., Honda, T., Kohr, G.: Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation. J. Math. Anal. Appl. 317, 302–319 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hamada, H., Honda, T.: Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. Chin. Ann. Math. 29(B(4)), 353–368 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Keogh, F.R., Merkes, E.P.: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 20, 8–12 (1969)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kohr, G.: Certain partial differential inequalities and applications for holomorphic mappings defined on the unit ball of \({\mathbb{C}}^n\). Ann. Univ. Mariae Curie Sklodowska. Sect. A 50, 87–94 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kohr, G.: On some best bounds for coefficients of several subclasses of biholomorphic mappings in \({\mathbb{C}}^n\). Complex Var. 36, 261–284 (1998)zbMATHGoogle Scholar
  16. 16.
    Kohr, G., Liczberski, P.: On strongly starlikeness of order alpha in several complex variables. Glas Mat. 33(53), 185–198 (1998)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kanas, S.: An unified approach to the Fekete-Szegö problem. Appl. Math. Comput. 218, 8453–8461 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    London, R.R.: Fekete-Szegö inequalities for close-to-convex functions. Proc. Am. Math. Soc. 117(4), 947–950 (1993)zbMATHGoogle Scholar
  19. 19.
    Liu, T.S., Ren, G.B.: The growth theorem for starlike mappings on bounded starlike circular domains. Chin. Ann Math. 19B, 401–408 (1998)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Liu, H., Lu, K.P.: Two subclasses of starlike mappings in several complex variables. Chin. Ann. Math. Ser.A. 21(5), 533–546 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Pfaltzgraff, J.A.: Subordination chains and univalence of holomorphic mappings in \({\mathbb{C}}^n\). Math. Ann. 210, 55–68 (1974)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pfaltzgraff, J.A.: Subordination chains and quasiconformal extension of holomorphic maps in \({\mathbb{C}}^n\). Ann. Acad. Sci. Fenn. Ser. A I Math. 1, 13–25 (1975)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Xu, Q.H., Liu, T.S.: On coefficient estimates for a class of holomorphic mappings. Sci. China Math. 52, 677–686 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Xu, Q.H., Liu, T.S., Liu, X.S.: The sharp estimates of homogeneous expansions for the generalized class of close-to-quasi-convex mappings. J. Math. Anal. Appl. 389, 781–791 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xu, Q.H., Liu, T.S., Zhang, W.J.: The Fekete and Szegö problem on bounded starlike circular domain in \({\mathbb{C}}^n\). Pure Appl. Math. Q. 12, 621–638 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Xu, Q.H., You, J.: Coefficient inequality for a subclass of biholomorphic mappings in several complex variables. Complex Var. Elliptic Equ. 63, 1306–1321 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Xu, Q.H., Liu, T.S., Liu, X.S.: On the Fekete and Szegö problem in one and higher dimensions. Sci. China Math. 61, 1775–1788 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Xu, Q.H., Xu, X.: On the coefficient inequality for a subclass of strongly starlike mappings of order \(\alpha \) in several complex variables. Results Math. (2018).  https://doi.org/10.1007/s00025-018-0837-2 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of ScienceZhejiang University of Science and TechnologyHangzhouPeople’s Republic of China

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