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The Lindelöf principle in ℂn

Abstract

Let D be a bounded domain in ℂn. A holomorphic function f: D → ℂ is called normal function if f satisfies a Lipschitz condition with respect to the Kobayashi metric on D and the spherical metric on the Riemann sphere ̅ℂ. We formulate and prove a few Lindelöf principles in the function theory of several complex variables.

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References

  1. Abate M., The Lindelöf principle and the angular derivative in strongly convex domains, J. Anal. Math., 1990, 54, 189–228

    MathSciNet  MATH  Article  Google Scholar 

  2. Abate M., Angular derivatives in strongly pseudoconvex domains, In: Several Complex Variables and Complex Geometry, 2, Santa Cruz, 1989, Proc. Sympos. Pure Math., 52(2), American Mathematical Society, Providence, 1991, 23–40

    Chapter  Google Scholar 

  3. Abate M., The Julia-Wolff-Carathéodory theorem in polydisks, J. Anal. Math., 1998, 74, 275–306

    MathSciNet  MATH  Article  Google Scholar 

  4. Abate M., Angular derivatives in several complex variables, In: Real Methods in Complex and CR Geometry, Lecture Notes in Math., 1848, Springer, Berlin, 2004, 1–47

    Book  Google Scholar 

  5. Abate M., Tauraso R., The Lindelöf principle and angular derivatives in convex domains of finite type, J. Aust. Math. Soc., 2002, 73(2), 221–250

    MathSciNet  MATH  Article  Google Scholar 

  6. Aladro G., Application of the Kobayashi metric to normal functions of several complex variables, Utilitas Math., 1987, 31, 13–24

    MathSciNet  MATH  Google Scholar 

  7. Aladro G., Krantz S.G., A criterion for normality in ℂn, J. Math. Anal. Appl., 1991, 161(1), 1–8

    MathSciNet  MATH  Article  Google Scholar 

  8. Bagemihl F., Seidel W., Sequential and continuous limits of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 1960, 280, 1–17

    MathSciNet  Google Scholar 

  9. Bayne R.E., Kwack M.H., A Lindelöf property for uniformly normal families, Missouri J. Math. Sci., 2010, 22(2), 130–138

    MathSciNet  MATH  Google Scholar 

  10. Cameron R.H., Storvick D.A., A Lindelöf theorem and analytic continuation for functions of several variables, with an application to the Feynman integral, In: Entire Functions and Related Parts of Analysis, LaJolla, 1966, American Mathematical Society, Providence, 1968, 149–156

    Chapter  Google Scholar 

  11. Cima J.A., Krantz S.G., The Lindelöf principle and normal functions of several complex variables, Duke Math. J., 1983, 50(1), 303–328

    MathSciNet  MATH  Article  Google Scholar 

  12. Čirka E.M., The Lindelöf and Fatou theorems in ℂn, Mat. Sb. (N.S.), 1973, 92(134), 622–644 (in Russian)

    MathSciNet  Google Scholar 

  13. Dovbush P.V., Normal functions of several complex variables, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1981, 36(1), 38–42 (in Russian)

    MathSciNet  Google Scholar 

  14. Dovbush P.V., Lindelöf’s theorem in ℂn, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1981, 36(6), 33–36 (in Russian)

    MathSciNet  Google Scholar 

  15. Dovbush P.V., Boundary behavior of normal holomorphic functions of several complex variables, Dokl. Akad. Nauk SSSR, 1982, 263(1), 14–17 (in Russian)

    MathSciNet  Google Scholar 

  16. Dovbush P.V., Lindelöf’s theorem in ℂn, Ukrainian Math. J., 1988, 40(6), 673–676

    MathSciNet  MATH  Article  Google Scholar 

  17. Dovbush P.V., Bloch functions on complex Banach manifolds, Math. Proc. R. Ir. Acad., 2008, 108(1), 27–32

    MathSciNet  Article  Google Scholar 

  18. Dovbush P.V., On normal and non-normal holomorphic functions on complex Banach manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2009, 8(1), 1–15

    MathSciNet  MATH  Google Scholar 

  19. Dovbush P.V., Boundary behaviour of Bloch functions and normal functions, Complex Var. Elliptic Equ., 2010, 55(1–3), 157–166

    MathSciNet  MATH  Google Scholar 

  20. Dovbush P.V., The Lindelöf principle for holomorphic functions of infinitely many variables, Complex Var. Elliptic Equ., 2011, 56(1–4), 315–323

    MathSciNet  MATH  Google Scholar 

  21. Dovbush P.V., On the Lindelöf-Gehring-Lohwater theorem, Complex Var. Elliptic Equ., 2011, 56(5), 417–421

    MathSciNet  MATH  Article  Google Scholar 

  22. Frosini C., Busemann functions and the Julia-Wolff-Carathéodory theorem for polydiscs, Adv. Geom., 2010, 10(3), 435–463

    MathSciNet  MATH  Article  Google Scholar 

  23. Funahashi K., Normal holomorphic mappings and classical theorems of function theory, Nagoya Math. J., 1984, 94, 89–104

    MathSciNet  MATH  Google Scholar 

  24. Garnett J.B., Marshall D.E., Harmonic Measure, New Math. Monogr., 2, Cambridge University Press, Cambridge, 2008

    MATH  Google Scholar 

  25. Gauthier P., A criterion for normalcy, Nagoya Math. J., 1968, 32, 277–282

    MathSciNet  MATH  Google Scholar 

  26. Gavrilov V.I., Dovbush P.V., Normal functions, Math. Montisnigri, 2001, 14, 5–61 (in Russian)

    MathSciNet  MATH  Google Scholar 

  27. Gehring F.W., Lohwater A.J., On the Lindelöf theorem, Math. Nachr., 1958, 19, 165–170

    MathSciNet  MATH  Article  Google Scholar 

  28. Hahn K.T., Inequality between the Bergman metric and Carathéodory differential metric, Proc. Amer. Math. Soc., 1978, 68(2), 193–194

    MathSciNet  MATH  Google Scholar 

  29. Hahn K.T., Asymptotic behavior of normal mappings of several complex variables, Canad. J. Math., 1984, 36(4), 718–746

    MathSciNet  MATH  Article  Google Scholar 

  30. Hahn K.T., Higher-dimensional generalizations of some classical theorems on normal meromorphic functions, Complex Variables Theory Appl., 1986, 6(2–4), 109–121

    MathSciNet  MATH  Article  Google Scholar 

  31. Hahn K.T., Nontangential limit theorems for normal mappings, Pacific J. Math., 1988, 135(1), 57–64

    MathSciNet  MATH  Article  Google Scholar 

  32. Järvi P., An extension theorem for normal functions, Proc. Amer. Math. Soc., 1988, 103(4), 1171–1174

    MathSciNet  MATH  Article  Google Scholar 

  33. Joseph J.E., Kwack M.H., Some classical theorems and families of normal maps in several complex variables, Complex Variables Theory Appl., 1996, 29(4), 343–362

    MathSciNet  MATH  Article  Google Scholar 

  34. Kobayashi S., Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan, 1967, 19(4), 460–480

    MathSciNet  MATH  Article  Google Scholar 

  35. Korányi A., Harmonic functions on Hermitian hyperbolic space, Trans. Amer. Math. Soc., 1969, 135, 507–516

    MathSciNet  MATH  Google Scholar 

  36. Krantz S.G., The Lindelöf principle in several complex variables, J. Math. Anal. Appl., 2007, 326(2), 1190–1198

    MathSciNet  MATH  Article  Google Scholar 

  37. Kwack M.H., Families of Normal Maps in Several Variables and Classical Theorems in Complex Analysis, Lecture Notes Ser., 33, Seoul National University, Seoul, 1996

    MATH  Google Scholar 

  38. Lehto O., Virtanen K.I., Boundary behaviour and normal meromorphic functions, Acta Math., 1957, 97(1–4), 47–65

    MathSciNet  MATH  Article  Google Scholar 

  39. Lindelöf E., Sur un Principe Général de l’Analyse et ses Applications á la Théorie de la Représentation Conforme, Acta Soc. Sci. Fennicae, 46(4), Suomen Tiedeseura, Helsinki, 1915

    Google Scholar 

  40. Montel P., Sur les familles de fonctions analytiques qui admettent des valeurs exceptionelles dans un domaine, Ann. Sci. École Norm. Sup., 1912, 29, 487–535

    MathSciNet  MATH  Google Scholar 

  41. Pommerenke Ch., Univalent Functions, Studia Mathematica/Mathematische Lehrbücher, 25, Vandenhoeck & Ruprecht, Göttingen, 1975

    Google Scholar 

  42. Sagan H., Space-Filling Curves, Universitext, Springer, New York, 1994

    MATH  Book  Google Scholar 

  43. Schiff J.L., Normal Families, Universitext, Springer, New York, 1993

    MATH  Book  Google Scholar 

  44. Stein E.M., Boundary Behavior of Holomorphic Functions of Several Complex Variables, Math. Notes, 11, Princeton University Press, Princeton, 1972

    MATH  Google Scholar 

  45. Whyburn G.T., Analytic Topology, Amer. Math. Soc. Colloq. Publ., 28, American Mathematical Society, New York, 1942

    MATH  Google Scholar 

  46. Zaidenberg M.G., Schottky-Landau growth estimates for s-normal families of holomorphic mappings, Math. Ann., 1992, 293(1), 123–141

    MathSciNet  Article  Google Scholar 

  47. Zavyalov B.I., Drozhzhinov Yu.N., On a multidimensional analogue of Lindelöf’s theorem, Dokl. Akad. Nauk SSSR, 1982, 262(2), 269–270 (in Russian)

    MathSciNet  Google Scholar 

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Correspondence to Peter V. Dovbush.

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Dovbush, P.V. The Lindelöf principle in ℂn . centr.eur.j.math. 11, 1763–1773 (2013). https://doi.org/10.2478/s11533-013-0274-0

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  • DOI: https://doi.org/10.2478/s11533-013-0274-0

MSC

  • 32A18

Keywords

  • Normal functions
  • Lindelöf principle
  • Admissible limits