Given two discrete random variables $X,Y$ such that $X,Y \in \mathbb{R}$ and $0 \leq X,Y \leq 1$, is it true that $$|\text{Cov}[X,Y] \leq \sqrt{\frac{1}{2} \text{I}[X,Y]}|. $$
This bound may be useful to my research if correct. What I have tried to do in order to prove it is the following:
1. Define the probability space $\mu$ over pairs of possible values of $X$ and $Y$: $$\Pr_\mu [(\alpha,\beta)] = \Pr[X = \alpha \text{ and } Y = \beta].$$
2. Similarly, let $\eta$ be the following probability space:
$$\Pr_\eta[(\alpha,\beta)] = \Pr[X=\alpha] \Pr[Y = \beta].$$
3. It can be seen that $D_{KL} (\mu ||\eta) = \text{I}[X,Y]$ where $D_{KL} (\mu ||\eta)$ is the Kullback–Leibler divergence (https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence).
4. Now, using Pinsker's inequality (https://en.wikipedia.org/wiki/Pinsker%27s_inequality) we get that: $$\delta(\mu,\eta) \leq \sqrt{\frac{1}{2} D_{KL}(\mu || \eta)} = \sqrt{\frac{1}{2}\text{I}[X,Y]}. $$
where $\delta()$ is the total variation distance .
5. The last step I tried is to find any connection between $\delta(\mu,\eta)$ and $|\text{Cov}[X,Y]|$ - ideally $|\text{Cov}[X,Y]| \leq \delta(\mu, \eta)$ - the problem is that seems like there is no such connection, I can find distribution where $|\text{Cov}[X,Y]| > \delta(\mu, \eta)$ and other distribution where the opposite holds.
I strongly believe that if this claim is correct, the proof goes through Pinsker's inequality, since it is very similar. Any advice will be appreciated.
EDIT:
As I was told in the comments, it holds that if $|X|, |Y| \leq 1$, $|\text{Cov}[X,Y]| \leq \delta (\mu, \eta)$ - the proof is in the comments. I was wrong when I thought I had a counter example. I also believe that the accepted answer below is correct.