1. Given that [tex]|\langle f|g\rangle|^2 \leqslant \langle f|f\rangle\langle g|g\rangle[/tex] prove that [tex]|\langle f|H|g\rangle|^2 \leqslant \langle f|H|f\rangle \langle g|H|g\rangle[/tex] where [tex]H[/tex] is a Hermitian and positive definite operator. 3. I tried to identify [tex]H|g\rangle[/tex] as a state and put it into the given inequality, but not so help. Is there any ideas to prove this inequality? Thanks in advance.