, Volume 15, Issue 1, pp 1–10 | Cite as

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces



We define nonnegative quasi-nearly subharmonic functions on so called locally uniformly homogeneous spaces. We point out that this function class is rather general. It includes quasi-nearly subharmonic (thus also subharmonic, quasisubharmonic and nearly subharmonic) functions on domains of Euclidean spaces \({{\mathbb{R}}^n}\), n ≥ 2. In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \({{\mathbb{R}}^n}\), n ≥ 2, are included. As examples we list the cases of the hyperbolic measure on the unit ball B n of \({{\mathbb{R}}^n}\), the \({{\mathcal{M}}}\)-invariant measure on the unit ball B 2n of \({{\mathbb{C}}^n}\), n ≥ 1, and the quasihyperbolic measure on any domain \({D\subset {\mathbb{R}}^n}\), \({D\ne {\mathbb{R}}^n}\). Moreover, we show that if u is a quasi-nearly subharmonic function on a locally uniformly homogeneous space and the space satisfies a mild additional condition, then also u p is quasi-nearly subharmonic for all p > 0.


Locally uniformly homogeneous space Hyperbolic measure \({{\mathcal{M}}}\)-invariant measure Quasihyperbolic measure Subharmonic Quasi-nearly subharmonic 

Mathematics Subject Classification (2000)

Primary 31B05 31C05 Secondary 31C45 


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© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

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

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