Product of $L^2$-norms
But is there a better bound for a general filter response?

No. The argument about it $H$ being a minimum phase filter doesn't make a difference. In essence you are asking a math question on some bound for
$$L = \sum X^2 \cdot \sum \frac{1}{X^2}$$
There is none. Let's assume a very simple signal $x = [a,1]$ . The product of the norms is
$$L = (a^2+1)(1/a^2+1) = a^2 + 2 + 1/a^2.$$
It's clear that $L &gt;=4$ and the equation is easily solvable for $a$ for any given value of $L$. In other words: by choosing $a$ you can make $L$ as large as you want it to be, so there is upper bound.