Matrix versions of some classical inequalities

Some natural inequalities related to rearrangement in matrix products can also be regarded as extensions of classical inequalities for sequences or integrals. In particular, we show matrix versions of Chebyshev and Kantorovich type inequalities. The matrix approach may also provide simplified proofs and new results for classical inequalities. For instance, we show a link between Cassel’s inequality and the basic rearrangement inequality for sequences of Hardy–Littlewood–Polya, and we state a reverse inequality to the Hardy–Littlewood–Polya inequality in which matrix technics are essential.