How can I prove that:
$$0<r^2<(r^2+(wl)^2)((1-w^2lc)^2+(wrc)^2)\tag1$$
$\forall c>0$ and all the other variables are bigger than zero using the scalar product of the vectors $A=(r,wl)$ and $B=(1-w^2lc,wrc)$?
I do not know how to get started on this problem and I do not see how I can use the scalar product of vectors to prove this inequality.
For $r\ne0$, $r^2>0$ is trivial. The other inequality is $(A\cdot B)^2<(A\cdot A)(B\cdot B)$, which is Cauchy–Schwarz for non-parallel vectors (otherwise we could only claim $\le$). In particular, check that $r(1-w^2lc)+wl(wrc)=r$. But $A$ could be parallel to $B$, so $<$ is in general incorrect.