  We establish Hardy--Littlewood inequalities for the Heckman--Opdam transform
associated to a general root datum $(\mathfrak{a},\Sigma,m)$ that generalizes
an analogous result for the spherical Fourier transform on a Riemannian
symmetric space of the non-compact type due to Eguchi and Kumahara. In
particular we obtain a more precise Hausdorff--Young inequality that
generalizes a recent result due to Narayanan, Pasquale, and Pusti.
