STRONG CONVEXITY FOR HARMONIC FUNCTIONS ON COMPACT SYMMETRIC SPACES

Let h be a harmonic function defined on a spherical disk. It is shown that Delta k|h|2 is nonnegative for all k is an element of N where Delta is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined on a disk in a normal homogeneous compact Riemannian manifold, and in particular in a symmetric space of the compact type. This complements a similar property for harmonic functions on Rn discovered by the first two authors and is related to strong convexity of the L2-growth function of harmonic functions.