Inequalities for surface integrals of non-negative subharmonic functions

Let H denote the class of positive harmonic functions on a bounded domain Ω in R . Let S be a sphere contained in Ω, and let σ denote the (N − 1)-dimensional measure. We give a condition on Ω which guarantees that there exists a constant K, depending only on Ω and S, such that R S u dσ ≤ K R ∂Ω udσ for every u ∈ H∩C(Ω). If this inequality holds for every such u, then it also holds for a large class of non-negative subharmonic functions. For certain types of domains explicit values for K are given. In particular the classical value K = 2 for convex domains is slightly improved.