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manuscripta mathematica

, Volume 131, Issue 3–4, pp 487–502 | Cite as

On some classes of bounded univalent mappings in several complex variables

  • Hidetaka Hamada
  • Gabriela Kohr
Article

Abstract

Let B n be the Euclidean unit ball in C n . In this paper, we study several properties of strongly starlike mappings of order α (0 < α < 1) and bounded convex mappings on B n . We prove that K-quasiregular strongly starlike mappings of order α on B n have continuous and univalent extensions to \({\overline{B}^n}\). We show that bounded convex mappings on B n are strongly starlike of some order α. We give a coefficient estimate for K-quasiregular strongly starlike mappings of order α on B n . Finally, we give examples of strongly starlike mappings of order α and bounded convex mappings on B n .

Mathematics Subject Classification (2000)

32H02 30C45 

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References

  1. 1.
    Bochner S.: Bloch’s theorem for real variables. Bull. Am. Math. Soc. 52, 715–719 (1946)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brannan D.A., Kirwan W.E.: On some classes of bounded univalent functions. J. Lond. Math. Soc.(2) 1, 431–443 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen H., Gauthier P.M.: Bloch constants in several variables. Trans. Am. Math. Soc. 353, 1371–1386 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chuaqui M.: Applications of subordination chains to starlike mappings in C n. Pac. J. Math. 168, 33–48 (1995)zbMATHMathSciNetGoogle Scholar
  5. 5.
    FitzGerald C.H., Thomas C.R.: Some bounds on convex mappings in several complex variables. Pac. J. Math. 165, 295–320 (1994)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Gong S.: Convex and Starlike Mappings in Several Complex Variables. Kluwer Acadamic Publisher, Dordrecht (1998)zbMATHGoogle Scholar
  7. 7.
    Graham I., Kohr G.: An extension theorem and subclasses of univalent mappings in several complex variables. Complex Var. Theory Appl. 47, 59–72 (2002)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Graham I., Kohr G.: Geometric Function Theory in One and Higher Dimensions. Marcel Dekker Inc., New York (2003)zbMATHGoogle Scholar
  9. 9.
    Graham I., Hamada H., Kohr G.: Parametric representation of univalent mappings in several complex variables. Can. J. Math. 54, 324–351 (2002)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Graham I., Hamada H., Kohr G., Suffridge T.J.: Extension operators for locally univalent mappings. Mich. Math. J. 50, 37–55 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Graham I., Hamada H., Kohr G.: Radius problems for holomorphic mappings on the unit ball in C n. Math. Nachr. 279, 1474–1490 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hallenbeck D.J., Ruscheweyh S.: Subordination by convex functions. Proc. Am. Math. Soc. 52, 191–195 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hamada H., Honda T.: Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. Chin. Ann. Math. B 29, 353–368 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hamada H., Kohr G.: Simple criterions for strongly starlikeness and starlikeness of certain order. Math. Nachr. 254–255, 165–171 (2003)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Hamada H., Kohr G.: Quasiconformal extension of biholomorphic mappings in several complex variables. J. Anal. Math. 96, 269–282 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hamada H., Honda T., Kohr G.: Parabolic starlike mappings in several complex variables. Manuscr. Math. 123, 301–324 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hörmander L.: On a theorem of Grace. Math. Scand. 2, 55–64 (1954)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Kato T.: Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19, 508–520 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kohr G.: On starlikeness and strongly starlikeness of order alpha in C n. Mathematica (Cluj) 40(63), 95–109 (1998)MathSciNetGoogle Scholar
  20. 20.
    Kohr G., Liczberski P.: On strongly starlikeness of order alpha in several complex variables. Glas. Math. III 33(53), 185–198 (1998)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Krzyz J.: Distortion theorems for bounded convex functions. Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 8, 625–627 (1960)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Liu, T.: The growth theorems, covering theorems and distortion theorems for biholomorphic mappings on classical domains. University of Science and Technology of China, Thesis (1989)Google Scholar
  23. 23.
    Liu H., Li X.: The growth theorem for strongly starlike mappings of order α on bounded starlike circular domains. Chin. Q. J. Math. 15, 28–33 (2000)Google Scholar
  24. 24.
    Marden A., Rickman S.: Holomorphic mappings of bounded distortion. Proc. Am. Math. Soc. 46, 226–228 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Mercer P.R.: A general Hopf lemma and proper holomorphic mappings between convex domains in C n. Proc. Am. Math. Soc. 119, 573–578 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Pfaltzgraff J.A.: Subordination chains and quasiconformal extension of holomorphic maps in C n. Ann. Acad. Sci. Fenn. Ser. AI Math. 1, 13–25 (1975)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Pfaltzgraff J.A., Suffridge T.J.: An extension theorem and linear invariant families generated by starlike maps. Ann. Univ. Mariae Curie Skl. 53, 193–207 (1999)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Poletsky E.A.: Holomorphic quasiregular mappings. Proc. Am. Math. Soc. 95, 235–241 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Pommerenke C.: On starlike and convex functions. J. Lond. Math. Soc. 37, 209–224 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Roper K., Suffridge T.J.: Convex mappings of the unit ball in C n. J. Anal. Math. 65, 333–347 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Roper K., Suffridge T.J.: Convexity properties of holomorphic mappings in C n. Trans. Am. Math. Soc. 351, 1803–1833 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Suffridge T.J.: The principle of subordination applied to functions of several variables. Pac. J. Math. 33, 241–248 (1970)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Suffridge T.J.: Biholomorphic mappings of the ball onto convex domains. Abstr. pap. Present. Am. Math. Soc. 11(66), 46 (1990)Google Scholar
  34. 34.
    Suffridge, T.J.: Holomorphic mappings of domains in C N onto convex domains. In: FitzGerald, C.H., Gong, S. (eds.) Geometric Function Theory in Several Complex Variables, pp. 295–309. World Scientific Publishing, River Edge (2004)Google Scholar
  35. 35.
    Väisälä, J.: Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics, vol. 229. Springer, Berlin, New York (1971)Google Scholar
  36. 36.
    Wu H.: Normal families of holomorphic mappings. Acta Math. 119, 193–233 (1967)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Faculty of EngineeringKyushu Sangyo UniversityHigashi-ku, FukuokaJapan
  2. 2.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

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