An inequality involving permanents of certain direct products

Abstract Let A denote a decomposable symmetric complex valued n-linear function on Cm. We prove ‖A·A‖ 2 ⩾2 n 2n n −1 ‖A⊗A‖ 2 , where · denotes the symmetric product and ⊗ the tensor product. As a consequence we have per M M M M ⩾2 n [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.