Homogeneous polynomials and extensions of Hardy‐Hilbert's inequality

Abstract

If L is a continuous symmetric n-linear form on a real or complex Hilbert space and L is the associated continuous n-homogeneous polynomial, then L = L . We give a simple proof of this well-known result, which works for both real and complex Hilbert spaces, by using a classical inequality due to S. Bernstein for trigonometric polynomials. As an application, an open problem for the optimal lower bound of the norm of a homogeneous polynomial, which is a product of linear forms, is related to the so-called permanent function of an n × n positive definite Hermitian matrix. We have also derived generalizations of Hardy-Hilbert's inequality.