Estimates for Convex Integral Means of Harmonic Functions

Abstract

AbstractWe prove that if f is an integrable function on the unit sphere S in ℝn, g is its symmetric decreasing rearrangement and u, v are the harmonic extensions of f, g in the unit ball , then v has larger convex integral means over each sphere rS, 0 < r < 1, than u has. We also prove that if u is harmonic in with |u| < 1 and u(0) = 0, then the convex integral mean of u on each sphere rS is dominated by that of U, which is the harmonic function with boundary values 1 on the right hemisphere and −1 on the left one.