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We investigate the possibility of improving the p -Poincaré inequality ‖ ∇ H N u ‖ p p ≥ Λ p ‖ u ‖ p p on the hyperbolic space, where p > 1 and Λ p : = [ ( N − 1 ) / p ] p is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincaré–Hardy inequality, namely an improvement of the best p -Poincaré inequality in terms of the Hardy weight r − p , r being geodesic distance from a given pole. Certain Hardy–Maz’ya-type inequalities in the Euclidean half-space are also obtained. [ABSTRACT FROM AUTHOR]