Homogeneous polynomials and extensions of Hardy‐Hilbert's inequality

If L is a continuous symmetric n‐linear form on a real or complex Hilbert space and $\widehat{L}$ is the associated continuous n‐homogeneous polynomial, then $\Vert L\Vert =\big \Vert \widehat{L}\big \Vert$. 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.