The Lindelöf principle in ℂn

Peter Dovbush

Open Mathematics (2013)

  • Volume: 11, Issue: 10, page 1763-1773
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Peter Dovbush. "The Lindelöf principle in ℂn." Open Mathematics 11.10 (2013): 1763-1773. <http://eudml.org/doc/269191>.

@article{PeterDovbush2013,
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.},
author = {Peter Dovbush},
journal = {Open Mathematics},
keywords = {Normal functions; Lindelöf principle; Admissible limits; normal functions; admissible limits},
language = {eng},
number = {10},
pages = {1763-1773},
title = {The Lindelöf principle in ℂn},
url = {http://eudml.org/doc/269191},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Peter Dovbush
TI - The Lindelöf principle in ℂn
JO - Open Mathematics
PY - 2013
VL - 11
IS - 10
SP - 1763
EP - 1773
AB - 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.
LA - eng
KW - Normal functions; Lindelöf principle; Admissible limits; normal functions; admissible limits
UR - http://eudml.org/doc/269191
ER -

References

top
  1. [1] Abate M., The Lindelöf principle and the angular derivative in strongly convex domains, J. Anal. Math., 1990, 54, 189–228 http://dx.doi.org/10.1007/BF02796148[Crossref] Zbl0694.32015
  2. [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 http://dx.doi.org/10.1090/pspum/052.2/1128532[Crossref] 
  3. [3] Abate M., The Julia-Wolff-Carathéodory theorem in polydisks, J. Anal. Math., 1998, 74, 275–306 http://dx.doi.org/10.1007/BF02819453[Crossref] Zbl0912.32005
  4. [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 http://dx.doi.org/10.1007/978-3-540-44487-9_1[Crossref] 
  5. [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 http://dx.doi.org/10.1017/S1446788700008818[Crossref] Zbl1113.32301
  6. [6] Aladro G., Application of the Kobayashi metric to normal functions of several complex variables, Utilitas Math., 1987, 31, 13–24 Zbl0585.32027
  7. [7] Aladro G., Krantz S.G., A criterion for normality in ℂn, J. Math. Anal. Appl., 1991, 161(1), 1–8 http://dx.doi.org/10.1016/0022-247X(91)90356-5[Crossref] 
  8. [8] Bagemihl F., Seidel W., Sequential and continuous limits of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 1960, 280, 1–17 Zbl0095.05801
  9. [9] Bayne R.E., Kwack M.H., A Lindelöf property for uniformly normal families, Missouri J. Math. Sci., 2010, 22(2), 130–138 Zbl1208.30033
  10. [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 http://dx.doi.org/10.1090/pspum/011/0237815[Crossref] 
  11. [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 http://dx.doi.org/10.1215/S0012-7094-83-05014-7[Crossref] Zbl0522.32003
  12. [12] Čirka E.M., The Lindelöf and Fatou theorems in ℂn, Mat. Sb. (N.S.), 1973, 92(134), 622–644 (in Russian) Zbl0285.32005
  13. [13] Dovbush P.V., Normal functions of several complex variables, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1981, 36(1), 38–42 (in Russian) Zbl0471.32002
  14. [14] Dovbush P.V., Lindelöf’s theorem in ℂn, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1981, 36(6), 33–36 (in Russian) 
  15. [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) Zbl0531.32003
  16. [16] Dovbush P.V., Lindelöf’s theorem in ℂn, Ukrainian Math. J., 1988, 40(6), 673–676 http://dx.doi.org/10.1007/BF01057192[Crossref] Zbl0679.32012
  17. [17] Dovbush P.V., Bloch functions on complex Banach manifolds, Math. Proc. R. Ir. Acad., 2008, 108(1), 27–32 http://dx.doi.org/10.3318/PRIA.2008.108.1.27[Crossref] Zbl1170.32001
  18. [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 Zbl1183.32004
  19. [19] Dovbush P.V., Boundary behaviour of Bloch functions and normal functions, Complex Var. Elliptic Equ., 2010, 55(1–3), 157–166 Zbl1192.32006
  20. [20] Dovbush P.V., The Lindelöf principle for holomorphic functions of infinitely many variables, Complex Var. Elliptic Equ., 2011, 56(1–4), 315–323 Zbl1256.32002
  21. [21] Dovbush P.V., On the Lindelöf-Gehring-Lohwater theorem, Complex Var. Elliptic Equ., 2011, 56(5), 417–421 http://dx.doi.org/10.1080/17476931003628240[Crossref] Zbl1219.32001
  22. [22] Frosini C., Busemann functions and the Julia-Wolff-Carathéodory theorem for polydiscs, Adv. Geom., 2010, 10(3), 435–463 http://dx.doi.org/10.1515/advgeom.2010.016[Crossref][WoS] 
  23. [23] Funahashi K., Normal holomorphic mappings and classical theorems of function theory, Nagoya Math. J., 1984, 94, 89–104 Zbl0533.32013
  24. [24] Garnett J.B., Marshall D.E., Harmonic Measure, New Math. Monogr., 2, Cambridge University Press, Cambridge, 2008 
  25. [25] Gauthier P., A criterion for normalcy, Nagoya Math. J., 1968, 32, 277–282 Zbl0157.39802
  26. [26] Gavrilov V.I., Dovbush P.V., Normal functions, Math. Montisnigri, 2001, 14, 5–61 (in Russian) 
  27. [27] Gehring F.W., Lohwater A.J., On the Lindelöf theorem, Math. Nachr., 1958, 19, 165–170 http://dx.doi.org/10.1002/mana.19580190111[Crossref] Zbl0089.05303
  28. [28] Hahn K.T., Inequality between the Bergman metric and Carathéodory differential metric, Proc. Amer. Math. Soc., 1978, 68(2), 193–194 Zbl0376.32020
  29. [29] Hahn K.T., Asymptotic behavior of normal mappings of several complex variables, Canad. J. Math., 1984, 36(4), 718–746 http://dx.doi.org/10.4153/CJM-1984-041-9[Crossref] Zbl0564.32015
  30. [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 http://dx.doi.org/10.1080/17476938608814163[Crossref] 
  31. [31] Hahn K.T., Nontangential limit theorems for normal mappings, Pacific J. Math., 1988, 135(1), 57–64 http://dx.doi.org/10.2140/pjm.1988.135.57[Crossref] Zbl0618.32004
  32. [32] Järvi P., An extension theorem for normal functions, Proc. Amer. Math. Soc., 1988, 103(4), 1171–1174 http://dx.doi.org/10.2307/2047105[Crossref] Zbl0659.32022
  33. [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 http://dx.doi.org/10.1080/17476939608814902[Crossref] 
  34. [34] Kobayashi S., Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan, 1967, 19(4), 460–480 http://dx.doi.org/10.2969/jmsj/01940460[Crossref] Zbl0158.33201
  35. [35] Korányi A., Harmonic functions on Hermitian hyperbolic space, Trans. Amer. Math. Soc., 1969, 135, 507–516 [Crossref] Zbl0174.38801
  36. [36] Krantz S.G., The Lindelöf principle in several complex variables, J. Math. Anal. Appl., 2007, 326(2), 1190–1198 http://dx.doi.org/10.1016/j.jmaa.2006.03.059[Crossref] 
  37. [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 
  38. [38] Lehto O., Virtanen K.I., Boundary behaviour and normal meromorphic functions, Acta Math., 1957, 97(1–4), 47–65 http://dx.doi.org/10.1007/BF02392392[Crossref] Zbl0077.07702
  39. [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 Zbl45.0665.02
  40. [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 Zbl43.0509.05
  41. [41] Pommerenke Ch., Univalent Functions, Studia Mathematica/Mathematische Lehrbücher, 25, Vandenhoeck & Ruprecht, Göttingen, 1975 
  42. [42] Sagan H., Space-Filling Curves, Universitext, Springer, New York, 1994 http://dx.doi.org/10.1007/978-1-4612-0871-6[Crossref] Zbl0806.01019
  43. [43] Schiff J.L., Normal Families, Universitext, Springer, New York, 1993 http://dx.doi.org/10.1007/978-1-4612-0907-2[Crossref] 
  44. [44] Stein E.M., Boundary Behavior of Holomorphic Functions of Several Complex Variables, Math. Notes, 11, Princeton University Press, Princeton, 1972 Zbl0242.32005
  45. [45] Whyburn G.T., Analytic Topology, Amer. Math. Soc. Colloq. Publ., 28, American Mathematical Society, New York, 1942 
  46. [46] Zaidenberg M.G., Schottky-Landau growth estimates for s-normal families of holomorphic mappings, Math. Ann., 1992, 293(1), 123–141 http://dx.doi.org/10.1007/BF01444708[Crossref] Zbl0766.32005
  47. [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) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.