Let $A, \, B \in \mathbb{R}^{n \times n}$ be $n \times n$, symmetric, positive definite matrices.
I would like to show that for any column vector $x \in \mathbb{R}^{n \times 1}$, it holds that $$ 0 \le \frac{(x^TA^{-1}x)^2}{(x^TB^{-1}x)(x^TA^{-1} B A^{-1}x) }\le 1.$$
In signal processing literature, if $A$ and $B$ are certain covariance matrices, then the fractional quantity I write above is sometimes referred to as the normalized signal to interference plus noise ratio. Showing this quantity is bounded below by zero is trivial, because of the positive definiteness of all the matrices involved. On the other hand, I probably need to use the Cauchy-Schwarz inequality in a clever way to show it bounded from above by one. This is where I am stuck. Hints or solutions are greatly appreciated.
Apply the Cauchy-Schwarz inequality, noticing that
$$x^T A^{-1} x = x^T B^{-1/2} B^{1/2} A^{-1}x = (B^{-1/2}x)^T B^{1/2} A^{-1}x.$$