2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123222In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold $M$. Furthermore, we also obtain sharp constant for the improved Hardy inequality and explicit constant for the Rellich inequality on hyperbolic space $\mathbb{H}^n$.Analysis of PDEsFunctional Analysis35J35; 26D10text