  This paper deals with the inequalities devoted to the comparison between the
norm of a function on a compact hypergroup and the norm of its Fourier
coefficients. We prove the classical Paley inequality in the setting of compact
hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley
(Pitt) inequalities in the noncommutative context. We establish H\"ormander's
$L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p \leq 2
\leq q<\infty$ as an application of Hausdorff-Young-Paley inequality. We
examine our results for the hypergroups constructed from the conjugacy classes
of compact Lie groups and for a class of countable compact hypergroups.
