An inequality for products of polynomials

Beauzamy, Bombieri, Enflo, and Montgomery recently established an inequality for the coefficients of products of homogeneous polynomials in several variables with complex coefficients (forms). We give this inequality an alternative interpretation: let /be a form of degree m, let f(D) denote the associated mth order differential operator, and define ∥f∥ by ∥f∥2 = f(D)f. Then ∥pq∥ ≥ ∥p∥ ∥q∥ for all forms p and q, regardless of degree or number of variables. Our principal result is that ∥pq∥ = ∥p∥ ∥q∥ if and only if, after a unitary change of variables, p and q are forms in disjoint sets of variables. This is achieved via an explicit formula for ∥pq∥ ² in terms of the coefficients of p and q.