Let W1Lp,q(Hn), 1≤q,p<∞ denote the Lorentz–Sobolev spaces of order one in the hyperbolic spaces Hn. Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in W1Lp,q(Hn) with 1≤q≤p which generalizes the result in [41] to the setting of Lorentz–Sobolev spaces. Second, we prove several sharp Poincaré–Sobolev type inequalities in W1Lp,q(Hn) with 1≤q≤p<n which generalize the results in [45] to the setting of Lorentz–Sobolev spaces. Finally, we provide the improved Moser–Trudinger type inequalities in W1Ln,q(Hn) in the critical case p=n with 1≤q≤n which generalize the results in [43] and improve the results in [57]. In the proof of the main results, we shall prove a Pólya–Szegö type principle in W1Lp,q(Hn) with 1≤q≤p which maybe is of independent interest.