Let λ1(A)⩾⋯⩾λn(A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽i1< ⋯ <ik⩽n. Our main results are

∑t=1kλt(GH)⩽∑t=1kλit(G)λn−it+1(H)

And

∑t=1kλit(GH)⩽∑t=1kλit(G)λn−t+1(H)

Here G and H are n by n positive semidefinite Hermitian matrices. These results extend Marshall and Olkin's inequality

∑t=1kλt(GH)⩽∑t=1kλt(G)λn−t+1(H)

We also present analogous results for singular values.