An Inequality of Hadamard Type for Permanents

Let F be an N×N complex matrix whose jth column is the vector ~ fj in C . Let |~ fj |2 denote the sum of the absolute squares of the entries of ~ fj . Hadamard’s inequality for determinants states that | det(F )| ≤ Nj=1 |~ fj |. Here we prove a sharp upper bound on the permanent of F , which is |perm(F )| ≤ N ! NN/2 N ∏ j=1 |~ fj |, and we determine all of the cases of equality. We also discuss the case in which |~ fj | is replaced by the lp norm of the vector ~ f considered as a function on {1, 2, . . . , N}. We note a simple sharp inequality for p = 1, and obtain bounds for intermediate p by interpolation. The interpolated bounds are not sharp, though there is a natural conjecture for what the sharp bounds should be. Mathemematics subject classification numbers: 15A45, 49M20