A Weighted Young Inequality and the Splitting of a Quadratic Form

We work in the setting of Euclidean Space and Real Matrices: Standing Notation, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, and Euclidean space $\mathbb{R}^{n}$ with its sum and difference of points, dot product and Euclidean norm $\lVert\,\cdot\,\rVert$, the real matrices, their sums and scalar multiples, the product, the matrix-vector product and the square $B^{2}=BB$, the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices and the norm $\lVert P\rVert$ of a symmetric real matrix, are all as fixed there. For $B\in\mathcal{S}(n)$ the matrix $B^{2}$ again lies in $\mathcal{S}(n)$, by Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix (ii).

Let $B\in\mathcal{S}(n)$ and let $\varepsilon\in\mathbb{R}$ be positive, so that $\varepsilon^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. Then the following hold.

1. (Weighted Young inequality) (i) For all $a,b\in\mathbb{R}^{n}$ and every positive $t\in\mathbb{R}$,

$$2\,(a\cdot b)\;\le\;t\,\lVert a\rVert^{2}+t^{-1}\lVert b\rVert^{2}.$$

2. (The perturbed matrix) (ii) $B+\varepsilon B^{2}\in\mathcal{S}(n)$.

3. (Splitting of a quadratic form) (iii) For all $x,z\in\mathbb{R}^{n}$,

$$x\cdot(Bx)\;\le\;z\cdot\bigl((B+\varepsilon B^{2})z\bigr)+\bigl(\varepsilon^{-1}+\lVert B\rVert\bigr)\,\lVert x-z\rVert^{2}.$$