I am trying to prove the following:
$$\langle\psi|\hat{H}|\phi\rangle\langle\phi|\hat{H}|\psi\rangle-\langle\psi|\hat{H}|\psi\rangle\langle\phi|\hat{H}|\phi\rangle\leqslant0.$$
I tried some ideas but could reach nowhere. I exploited the fact that $\hat{H}$ is hermitian, and thus the first term in the inequality became $\langle\phi|\hat{H}|\psi\rangle^*\langle\phi|\hat{H}|\psi\rangle=|\langle\phi|\hat{H}|\psi\rangle|^2$ and then by Cauchy Schwartz inequality, $|\langle\phi|\hat{H}|\psi\rangle|^2\leqslant\langle\phi|\phi\rangle\langle\psi|\hat{H}\hat{H}|\psi\rangle$,but I can see this just removes $|\phi\rangle$ from the game.
Another idea was to write $\hat{H}$ in the first term in outer product notation, this gives:$$\langle\psi|\hat{H}|\phi\rangle\langle\phi|\hat{H}|\psi\rangle=\sum_{i,j}E_iE_j\langle\psi|i\rangle\langle i|\phi\rangle\langle\phi|j\rangle\langle j|\psi\rangle$$I tried to work on this to get the inequality, but it got me nowhere.
Any help is appreciated.