  We first prove Kato's inequalities for the Laplacian and a
Schr$\ddot{o}$dinger-type operator on smooth functions on closed Riemannian
manifolds. We then apply the result to establish some new domination
inequalities for spectral zeta functions and their related spectral zeta
kernels on $n$-dimensional unit spheres using Kato's inequalities and
majorisation techniques. Our results are the generalisations of Kato's
comparison inequalities for Riemannian surfaces to $n$-dimensional closed
Riemannian manifolds.
