Lower bounds for the norm of the symmetric product

Suppose each of m, n, and k is a positive integer, k ? n, A is a (real-valued) symmetric n-linear function on E_m, and B is a k-linear symmetric function on E_m. The tensor and symmetric products of A and B are denoted, respectively, by A ?B and A?B. The identity

$\text{6A · B6}{}^2\text{=}\text{∑}\text{q=0}\text{n}\text{(}\text{n}\text{k}\text{)}\text{(}\text{n+k}\text{k}\text{)}\text{6A?}{}_{\mathrm{q}}\text{B6}{}^2$

is proven by Neuberger in [1]. An immediate consequence of this identity is the inequality

$\text{6A · B 6}{}^2\text{?}\text{n+k}\text{n}{}^{\mathrm{?1}}\text{6A · B 6}{}^2$

In this paper a necessary and sufficient condition for

$\text{6A · B 6}{}^2\text{=}\text{n+k}\text{n}{}^{\mathrm{?}}\text{6A · B 6}{}^2$

is given. It is also shown that under certain conditions the inequality can be considerably improved. This improvement results from an analysis of the terms 6A?_qB6, 1?q?n, appearing in the identity.