Mean value property and a Berezin-type transform on the half-space

Let B be the open unit ball in R n . Liu (2007) [6] has shown that if f ∈ C ( B ¯ ) then the iterates of a Berezin-type transform B B k f of f converge to the Poisson extension of the boundary values of f, as k → ∞ . In this paper, we extend this to the half-space setting. First, we obtain the mean value property for harmonic functions on the half-space H. Based on this property, we define a Berezin-type transform B H and investigate the limit of the iterates of B H .