PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

The real and complex convexity

Autorzy
Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We prove that the holomorphic differential equation ϕ’’(ϕ+c) = γ(ϕ’)² (ϕ:C→C be a holomorphic function and (γ, c) ϵ C²) plays a classical role on many problems of real and complex convexity. The condition exactly γ ϵ [wzór] (independently of the constant c) is of great importance in this paper. On the other hand, let n ≥ 1, (A₁, A₂) ϵ C² and g₁, g₂ : Cᵑ → C be two analytic functions. Put u(z, w) = │A ₁w - g₁(z) │² + │A₂w - g₂(z) │²v(z,w) = │A₁w - g₁(z) │² + │ A₂w - g₂(z) │², for (z,w) ϵ Cᵑ x C. We prove that u is strictly plurisubharmonic and convex on Cᵑ x C if and only if n = 1, (A₁, A₂) ϵ C² \{0} and the functions g₁ and g₂ have a classical representation form described in the present paper. Now v is convex and strictly psh on Cᵑ x C if and only if (A₁, A₂) ϵ C² \{0}, n ϵ {1,2} and and g₁, g₂ have several representations investigated in this paper.
Rocznik
Tom
Strony
123--156
Opis fizyczny
Bibliogr. 35 poz.
Twórcy
autor
  • Département de Mathématiques, Faculté des Sciences de Tunis, 1060 Tunis, Tunisia, abidijamel1@gmail.com
Bibliografia
  • 1. J. Abidi, Sur quelques problèmes concernant les fonctions holomorphes et plurisousharmoniques, Rend. Circ. Mat. Palermo 51 (2002) 411-424.
  • 2. J. Abidi, M.L. Ben Yattou, Le minimum de deux fonctions plurisousharmoniques et une nouvelle caractérisation des fonctions holomorphes, Math. Bohem. 136 (2011) 301-310.
  • 3. J. Abidi, Contribution à l'étude des fonctions plurisousharmoniques convexes et analytiques, Serdica Math. J. 40 (2014) 329-388.
  • 4. S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer-Verlag New York, 1992.
  • 5. C.A. Berenstein, R. Gay, Complex Variables. An Introduction, Graduate Texts In Mathematics 125, Springer Verlag, New York, 1991.
  • 6. U. Cegrell, Removable singularities for plurisubharmonic functions and related problems, Proc. Lond. Math. Soc. 36 (1978) 310-336.
  • 7. U. Cegrell, Removable singularity sets for analytic functions having modulus with bounded Laplace mass, Proc. Amer. Math. Soc. 88 (1983) 283-286.
  • 8. U. Cegrell, L. Hed, Subextension and approximation of negative plurisubharmonic functions, Michigan Math. J. 56 (2008) 593-601.
  • 9. D. Coman, V. Guedj, A. Zeriahi, Extension of plurisubharmonic functions with growth control, J. Reine Angew. Math. 676 (2013) 33-49.
  • 10. J.B. Conway, Functions of One Complex Variable II, Springer-Verlag, 1995.
  • 11. A.Edigarian, J. Wiegerinck, The pluripolar hull of the graph of a holomorphic function with polar singularities, Indiana Univ. Math. J. 52 (2003) 1663-1680.
  • 12. A. Edigarian, J. Wiegerinck, Determination of the pluripolar hull of graphs of certain holomorphic functions, Ann. Inst. Fourier, Grenoble 54 (2004) 2085-2104.
  • 13. R.C. Gunning, H. Rossi, Analytic Functions of Several Complex Variables, Prentice - Hall, Englewood Cliffs, 1965.
  • 14. G.M. Henkin, J. Leiterer, Theory of Functions on Complex Manifolds, Birkhäuser, Boston, Mass., 1984.
  • 15. M. Hervé, Les Fonctions Analytiques, Presses Universitaires de France, 1982.
  • 16. L. Hörmander, An Introduction to Complex Analysis in Several Variables, Third Edition (revised), Mathematical Library, Vol. 7, North Holland, Amsterdam-New York-Oxford-Tokyo, 1990.
  • 17. L. Hörmander, Notions of Convexity, Birkhäuser, Basel-Boston-Berlin, 1994.
  • 18. J. Hyvönen, J. Rühentaus, On the extension in the Hardy classes and in the Nevanlinna class, Bull. Soc. Math. France 112 (1984) 469-480.
  • 19. M. Jarnicki, P. Pug, Extension of Holomorphic Functions, de Gruyter, Berlin, 2000.
  • 20. M. Klimek, Pluripotential Theory, Clarendon Press, Oxford, 1991.
  • 21. S.G. Krantz, Function Theory of Several Complex Variables, Wiley, New York, 1982.
  • 22. F. Lárusson, R. Sigurdsson, Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math. 501 (1998) 1-39.
  • 23. F. Lárusson, R. Sigurdsson, Siciak-Zahariuta extremal functions, analytic discs and polynomial hulls, Math. Ann. 345 (2009) 159-174.
  • 24. P. Lelong, Fonctions Plurisousharmoniques et Formes Différentielles Positives, Gordon et Breach, New-York et Dunod, Paris, 1969.
  • 25. P. Lelong, Définition des fonctions plurisousharmoniques, C. R. Acad. Sci. Paris 215 (1942) 398-400.
  • 26. P. Lelong, Sur les suites des fonctions plurisousharmoniques, C. R. Acad. Sci. Paris 215 (1942) 454-456.
  • 27. P. Lelong, Les fonctions plurisousharmoniques, Ann. Sci. Ecole Norm. Sup. 62 (1945) 301-338.
  • 28. E. Poletsky, Holomorphic currents, Indiana Univ. Math. J. 42 (1993) 85-144.
  • 29. E. Poletsky, The minimum principle, Indiana Univ. Math. J. 51 (2002) 269-303.
  • 30. R.M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer-Verlag, 1986.
  • 31. J. Riihentaus, On the extension of separately hyperharmonic functions and Hp-functions, Michigan Math. J. 31 (1984) 99-112.
  • 32. L.I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables, Amer. Math. Soc., Providence, RI 1974.
  • 33. W. Rudin, Function Theory in Polydiscs, Benjamin, New York, 1969.
  • 34. W. Rudin, Function Theory in the Unit Ball of Cn, Springer-Verlag, New York, 1980.
  • 35. V.S. Vladimirov, Les fonctions de plusieurs variables complexes (et leur application à la théorie quantique des champs), Paris: Dunod, 1967.
Uwagi
PL
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-84f2972f-deec-41e5-951d-107c7be534f8
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.