I am looking for an upper bound on the following quantity \begin{align*} A=E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right] \end{align*} where $Z$ is standard Gaussian and $Y=Z+W$ where $Z$ and $W$ are independent and $E[W^2]\le 1$.
Here is some bounds I was able to come up with:
Bound 1 \begin{align*} A=E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right] &\le E \left[\left(E[Z^2\mid Y] \right)^2\right]\\ & \le E[Z^4] =3 \end{align*}
Bound 2 \begin{align*} A=E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right] &\le E \left[\left(E[Z^2\mid Y] \right)^2\right]- E \left[ |E[Z\mid Y]| |E[Z\mid Y]| \right] \\ &=E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ (E[Z\mid Y])^2 \right]\\ &=3- 2E \left[ (E[Z\mid Y])^2 \right] \end{align*}
Note, that having about in terms of $E \left[ (E[Z\mid Y])^2 \right]$ is acceptable for me. My question can we push bound in 2 even lower ??
I am also looking for any suggestion or tricks that you might have that can help with bounding.
Thank you