Let $H $ be a hermitian matrix in complex hilbert with $\Vert H \Vert = K$. I am trying to show the following for any unit vector $v \in S$ and $w \in T$ where $S,T$ are orthogonal, we have $-r_2^2 v^*Hv+2r_1r_2Real(v^*Hw)+r_2^2w^*Hw \geq -2K r_2^2-2Kr_2$.
Edit: $r_1,r_2 $ are real numbers smaller than 1.