Abstract :
Given positive integers n and p, and a complex finite dimensional vector space V, we let Sn,p(V) denote the set of all functions from V×V×cdots, three dots, centered×V-(n+p copies) to image that are linear and symmetric in the first n positions, and conjugate linear symmetric in the last p positions. Letting κ=min{n,p} we introduce twisted inner products, [·,·]s,t,1less-than-or-equals, slants,tless-than-or-equals, slantκ, on Sn,p(V), and prove the monotonicity condition [F,F]s,tgreater-or-equal, slanted[F,F]u,v is satisfied when sless-than-or-equals, slantuless-than-or-equals, slantκ,tless-than-or-equals, slantvless-than-or-equals, slantκ, and Fset membership, variantSn,p(V). Using the monotonicity condition, and the Cauchy-Schwartz inequality, we obtain as corollaries many known inequalities involving norms of symmetric multilinear functions, which in turn imply known inequalities involving permanents of positive semidefinite Hermitian matrices. New tensor and permanental inequalities are also presented. Applications to partial differential equations are indicated.
