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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{L}$\end{document} is the associated continuous n ‐homogeneous polynomial, then \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Vert L\Vert =\big \Vert \widehat{L}\big \Vert$\end{document} . 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.