Help with a inequality

Question

I need to understand this inequality and show if $x^t H x < M ||x||^2$ then $x^t H^t H x < M^2 ||x||^2$ with H is positive definite and $M>0$, in general.

Answer 1

$$x^t H^t H x = (x^t H^t \frac{x x^t}{||x||^2} H x) = \frac{1}{||x||^2} (x^t H x )^t ( x^t H x) \le \frac{1}{||x||^2} (M||x||^2)^t (M||x||^2) = M^2||x||^2 $$