Strong convexity for harmonic functions on compact symmetric spaces

Let $h$ be a harmonic function defined on a spherical disk. It is shown that ${\mathrm{\Delta }}^k\mathrm{\mid }h{\mathrm{\mid }}^2$ is nonnegative for all $k\in \mathbb{N}$ where $\mathrm{\Delta }$ 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 ${\mathbb{R}}^n$ discovered by the first two authors and is related to strong convexity of the $L^2$-growth function of harmonic functions.