Need help with inequality to understand stability analysis

Question

I have this

$$ | \xi |^{2} = 1 - 4p^{2}(1-p^{2})s^{4}$$

where $s = \sin\left(\frac\omega 2 \right)$. The method is said to be stable if $ | \xi|\leq1$.

From here I am supposed to deduce that this scheme is stable for $ -1 \leq p \leq 1$, but I do not know where to go from here... Can anyone suggest how?

Answer 1

Hints:

1. If $p^2\in[0,1]$ then $4p^2(1-p^2)\in [0, 1]$, say by AM-GM.

2. If $p^2>1$, then $(1-p^2)$ is negative, so your LHS is $>1$ unless $s=0$.

Answer 2

Is this a correct analysis?

$ 1-4p^{2}(1-p^{2}) s^{4} \leq 1 $

$ -4p^{2}(1-p^{2}) s^{4} \leq 0 $

$ p^{2}(1-p^{2}) \geq 0$ Since sine is bounded between 0 and 1

$p^{2} \geq 0 $ is always true. So $ 1-p^{2} \geq 0 $ gives $p^{2} \leq 1 $