An inequality involving permanents of certain direct products

AbstractLet A denote a decomposable symmetric complex valued n-linear function on Cm. We prove ‖A·A‖2⩾2n2nn−1‖A⊗A‖2, where · denotes the symmetric product and ⊗ the tensor product. As a consequence we have per MMMM⩾2n[per(M)]2, where M is a positive semidefinite Hermitian matrix and per denotes the permanent function. A sufficient condition for equality in the matrix inequality is that M is a nonnegative diagonal matrix