Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the squares of the entries of $\mathbf{v}$ and $\mathbf{w}$.
My question is how to prove that
\begin{align*}
\|\mathbf{v}^2\|\|\mathbf{w}^2\| – \langle \mathbf{v}^2,\mathbf{w}^2\rangle \leq \|\mathbf{v}\|^2\|\mathbf{w}\|^2 – \langle \mathbf{v},\mathbf{w}\rangle^2.
\end{align*}
Some notes are in order:
- The Cauchy-Schwarz inequality tells us that both sides of this inequality are non-negative. Thus the proposed inequality is a strengthening of Cauchy-Schwarz that gives a non-zero bound on the RHS.
- I know that this inequality is true, but my method of proving it is extremely long and roundabout. It seems like it should have a straightforward-ish proof, or should follow from another well-known inequality, and that's what I'm looking for.
Best Answer
Here is a proof for every $n$. Using the notation $\mathbf{v}=(v_1,\dots,v_n)$ and $\mathbf{w}=(w_1,\dots,w_n)$, the inequality reads $$\left(\sum_i v_i^4\right)^{1/2}\left(\sum_i w_i^4\right)^{1/2}-\sum_i v_i^2 w_i^2\leq \left(\sum_i v_i^2\right)\left(\sum_i w_i^2\right)-\left(\sum_i v_i w_i\right)^2.$$ Rewriting the right hand side in a familiar way, and then rearranging and squaring, we obtain the equivalent form $$\left(\sum_i v_i^4\right)\left(\sum_i w_i^4\right)\leq\left(\sum_i v_i^2 w_i^2+\sum_{i<j}(v_iw_j-v_j w_i)^2\right)^2.$$ Rewriting the left hand side in a familiar way, we obtain the equivalent form $$\left(\sum_i v_i^2w_i^2\right)^2+\sum_{i<j}(v_i^2w_j^2-v_j^2w_i^2)^2\leq\left(\sum_i v_i^2 w_i^2+\sum_{i<j}(v_iw_j-v_j w_i)^2\right)^2.$$ Equivalently, $$\sum_{i<j}(v_i^2w_j^2-v_j^2w_i^2)^2\leq 2\left(\sum_k v_k^2w_k^2\right)\sum_{i<j}(v_iw_j-v_j w_i)^2+\left(\sum_{i<j}(v_iw_j-v_j w_i)^2\right)^2.$$ It will be clear in a moment why we renamed the variable $i$ to $k$ in the first sum on the right hand side. Namely, we claim that the following stronger inequality holds: $$\sum_{i<j}(v_i^2w_j^2-v_j^2w_i^2)^2\leq 2\sum_{i<j}(v_i^2w_i^2+v_j^2w_j^2)(v_iw_j-v_j w_i)^2+\sum_{i<j}(v_iw_j-v_j w_i)^4.$$ Indeed, this inequality can be rearranged to $$0\leq 2\sum_{i<j}(v_iw_i-v_jw_j)^2(v_iw_j-v_jw_i)^2,$$ and we are done.