Potential Analysis

, Volume 29, Issue 1, pp 89–104 | Cite as

Classes of Quasi-nearly Subharmonic Functions

  • Miroslav Pavlović
  • Juhani Riihentaus


It is well known and important that if u ≥ 0 is subharmonic on a domain Ω in ℝ n and p > 0, then there is a constant C(n,p) ≥ 1 such that \(u(x)^p\leq C(n,p){\mathcal{MV}}(u^p,B(x,r))\) for each open ball B(x,r) ⊂ Ω. The definition of a relatively new function class, quasi-nearly subharmonic functions, is based on such a generalized mean value inequality. It is pointed out that the obtained function class is natural. It has important and interesting properties and, at the same time, it is large: In addition to nonnegative subharmonic functions, it includes, among others, Hervé’s nearly subharmonic functions, functions satisfying certain natural growth conditions, especially certain eigenfunctions, polyharmonic functions and generalizations of convex functions. Further, some of the basic properties of quasi-nearly subharmonic functions are stated in a unified form. Moreover, a characterization of quasi-nearly subharmonic functions with the aid of the quasihyperbolic metric and two weighted boundary limit results are given.


Subharmonic Quasi-nearly subharmonic Bochner-Martinelli formula Approach region Boundary limit 

Mathematics Subject Classifications (2000)

Primary 31C05 31B25 Secondary 31B05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahern, P., Bruna, J.: Maximal and area integral characterizations of Hardy-Sobolev spaces in the unit ball of ℂn. Rev. Mat. Iberoam. 4, 123–153 (1988)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Di Benedetto, E., Trudinger, N.S.: Harnack inequalities for quasi-minima of variational integrals. Ann. Inst. Henri Poincare Anal. Non Lineaire 1, 295–308 (1984)Google Scholar
  3. 3.
    Domar, Y.: Uniform boundedness in families related to subharmonic functions. J. Lond. Math. Soc. 38(2), 485–491 (1988)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dyakonov, K.M.: Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math. 178, 143–167 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Dyakonov, K.M.: Holomorphic functions and quasiconformal mappings with smooth moduli. Adv. Math. 187, 146–172 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Fefferman, C., Stein, E.M.: Hp spaces of several variables. Acta Math. 129, 137–193 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Garnett, J.B.: Bounded Analytic Functions. Academic, New York (1981)zbMATHGoogle Scholar
  8. 8.
    Hallenbeck, D.J.: Radial growth of subharmonic functions. Pitman Res. Notes 262, 113–121 (1992)MathSciNetGoogle Scholar
  9. 9.
    Hervé, M.: Analytic and Plurisubharmonic Functions in Finite and Infinite Dimensional Spaces. Lecture Notes in Mathematics 198. Springer, Berlin (1971)zbMATHGoogle Scholar
  10. 10.
    Jevtić, M., Pavlović, M.: \({\mathcal{M}}\)-Besov p-classes and Hankel operators in the Bergman spaces on the unit ball. Arch. Math. 61, 367–376 (1993)CrossRefzbMATHGoogle Scholar
  11. 11.
    Jevtić, M., Pavlović, M.: Subharmonic behavior of generalized (α,β)-harmonic functions and their derivatives. Indian J. Pure Appl. Math. 30, 407–418 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Krasnoselskiĭ, M.A., Rutickiĭ, Ja.B.: Convex functions and Orlicz spaces. Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen (1961)Google Scholar
  13. 13.
    Kuran, Ü.: Subharmonic behavior of |h|p (p > 0, h harmonic). J. Lond. Math. Soc. 8(2), 529–538 (1974)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Mateljević, M., Pavlović, M.: \({\mathcal{L}}^p\)-behavior of power series with positive coefficients and Hardy spaces. Proc. Am. Math. Soc. 87, 309–316 (1983)CrossRefzbMATHGoogle Scholar
  15. 15.
    Mizuta, Y.: Potential theory in euclidean spaces. Gaguto International Series, Mathematical Sciences and Applications, 6. Gakkōtosho Co., Tokyo (1996)Google Scholar
  16. 16.
    Mizuta, Y.: Boundary limits of functions in weighted Lebesgue or Sobolev classes. Rev. Roum. Math. Pures Appl. 46, 67–75 (2001)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pavlović, M.: Mean values of harmonic congugates in the unit disc. Complex Var. Theory Appl. 10, 53–65 (1988)zbMATHGoogle Scholar
  18. 18.
    Pavlović, M.: Inequalities for the gradient of eigenfunctions of the invariant Laplacian on the unit ball. Indag. Math. (N.S.) 2, 89–98 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Pavlović, M.: A proof of the Hardy-Littlewood theorem on fractional integration and a generalization. Publ. Inst. Math. (Belgrade) 59, 31–38 (1996)Google Scholar
  20. 20.
    Pavlović, M.: On subharmonic behavior and oscillation of functions on balls in ℝn. Publ. Inst. Math. (Beograd) 55(69), 18–22 (1994)MathSciNetGoogle Scholar
  21. 21.
    Pavlović, M.: Subharmonic behavior of smooth functions. Mat. Vesn. 48, 15–21 (1996)zbMATHGoogle Scholar
  22. 22.
    Pavlović, M.: Decompositions of \({\mathcal{L}}^p\) and Hardy spaces of polyharmonic functions. J. Math. Anal. Appl. 216, 499–509 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Pavlović, M.: On Dyakonov’s paper ‘Equivalent norms on Lipschitz-type spaces of holomorphic functions’. Acta Math. 183, 141–143 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Pavlović, M.: Introduction to Function Spaces on the Disk. Posebna Izdanja 20, Matematic̆ki Institut SANU (2004)Google Scholar
  25. 25.
    Pavlović, M.: Lipschitz conditions on the modulus of a harmonic function. Rev. Mat. Iberoam. 23, 831–845 (2007)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Riihentaus, J.: On a theorem of Avanissian–Arsove. Expo. Math. 7, 69–72 (1989)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Riihentaus, J.: Subharmonic functions: Non-tangential and tangential boundary behavior. In: Mustonen, V., Rákosnik, J. (eds.) Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA’99), Proceedings of the Syöte Conference 1999, pp. 229–238. Math. Inst., Czech Acad. Science, Praha, (2000) (ISBN 80-85823-42-X)Google Scholar
  28. 28.
    Riihentaus, J.: A generalized mean value inequality for subharmonic functions. Expo. Math. 19, 187–190 (2001)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Riihentaus, J.: A generalized mean value inequality for subharmonic functions and applications. arXiv:math.CA/0302261 v1, 21 Feb (2003)Google Scholar
  30. 30.
    Riihentaus, J.: A weighted boundary behavior result for subharmonic functions. Adv. Algebra Anal. 1, 27–38 (2006)MathSciNetGoogle Scholar
  31. 31.
    Rudin, W.: Function Theory in the Unit Ball of ℂn. Springer, New York (1980)Google Scholar
  32. 32.
    Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables I. The theory of Hp spaces. Acta Math. 103, 25–62 (1960)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Stoll, M.: Invariant Potential Theory in the Unit Ball of ℂn. London Mathematical Society Lecture Notes Series, Cambridge (1994)Google Scholar
  34. 34.
    Stoll, M.: Boundary limits and non-integrability of \({\mathcal{M}}\)-subharmonic functions in the unit ball of ℂn (n ≥ 1). Trans. Am. Math. Soc. 349, 3773–3785 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Stoll, M.: Weighted tangential boundary limits of subharmonic functions on domains in ℝn (n ≥ 2). Math. Scand. 83, 300–308 (1998)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Suzuki, N.: Nonintegrability of harmonic functions in a domain. Jpn. J. Math. 16, 269–278 (1990)zbMATHGoogle Scholar
  37. 37.
    Suzuki, N.: Nonintegrability of superharmonic functions. Proc. Am. Math. Soc. 113, 113–115 (1991)CrossRefzbMATHGoogle Scholar
  38. 38.
    Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Academic, London (1986)zbMATHGoogle Scholar
  39. 39.
    Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics 1319, Springer, Berlin (1988)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Matematic̆ki FakultetBelgradeSerbia
  2. 2.Department of MathematicsUniversity of JoensuuJoensuuFinland

Personalised recommendations