An Invariant Volume-Mean-Value Property

If ƒ is harmonic and integrable over the open unit disc U then so is ƒ ∘ ψ for every Moebius transformation ψ of U, and therefore 1π ∫ U (ƒ ∘ ψ) d A = ƒ(ψ(0) for every ψ. Conversely, does this mean-value property imply that ƒ is harmonic? A more general question, with the unit ball Bn of C (for arbitrary n≥ 1) in place of the disc, is investigated in the present paper. The answer is found to be affirmative if n ≤ 11, negative if n ≤ 12.