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 R-n, n >= 2. In addition, quasi-nearly subharmonic functions with respect to various measures on domains of R-n, n >= 2, are included. As examples we list the cases of the hyperbolic measure on the unit ball B-n of R-n, the M-invariant measure on the unit ball B-2n of C-n, n >= 1, and the quasihyperbolic measure on any domain D subset of R-n, D not equal 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.