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

Let B be the open unit ball in Rn. Liu (2007) [6] has shown that if then the iterates of a Berezin-type transform 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 BH and investigate the limit of the iterates of BH.