For any couple of random variables $X$ and $Y$, $$ \displaystyle |\mathbb{E} (XY)|^{2}\leq \mathbb{E} (X^{2})\mathbb{E} (Y^{2}) $$ Is it possible to put an intermediate bound, like $$ \displaystyle |\mathbb{E} (XY)|^{2}\leq\color{red}{ \mathbb{E} (|X|)\mathbb{E} (Y^{2})}\leq \mathbb{E} (X^{2})\mathbb{E} (Y^{2}) $$ ?
Addendum after the answer given by @Prem. What I'm really looking for is a bound of this form $$ \mathbb{E}(XY)\leq \mathbb{E}[|X|]\bigg(\text{whatever that wouldn't contain } \frac{1}{\mathbb{E}[|X|]}\bigg) $$
The Question title says that you want "something" between the 2 terms , while the Body gives only some sort of Example term. [[ User "coffeemath" has already highlighted that this Example term will not work ]]
$$\displaystyle |\mathbb{E} (XY)|^{2} \leq [[\ \ ??\ \ ]] \leq \mathbb{E} (X^{2})\mathbb{E} (Y^{2})$$
Well , there are various ways we can go about that. I will give only 2 Examples.
(1) Using Averaging :
$$\displaystyle |\mathbb{E} (XY)|^{2} \leq [[\ \ (|\mathbb{E} (XY)|^{2} + \mathbb{E} (X^{2})\mathbb{E} (Y^{2}))/2\ \ ]] \leq \mathbb{E} (X^{2})\mathbb{E} (Y^{2})$$
Here , we are putting the Average between the 2 values , hence it will be true.
(2) Using Expectation of Absolute value :
$$\displaystyle |\mathbb{E} (XY)|^{2} \leq [[\ \ |\mathbb{E} (|XY|)|^{2}\ \ ]] \leq \mathbb{E} (X^{2})\mathbb{E} (Y^{2})$$
Here , we are converting $XY$ to $|XY|$ before taking Expectation , hence it will be between the other 2 terms.
There are other ways we can put "something" between the 2 terms.