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Potential Analysis

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

Classes of Quasi-nearly Subharmonic Functions

  • Miroslav Pavlović
  • Juhani Riihentaus
Article

Abstract

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.

Keywords

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

Mathematics Subject Classifications (2000)

Primary 31C05 31B25 Secondary 31B05 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

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

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