I have two random variables $X$ and $Y$ (defined on the same probability space) both with point masses at zero:
$$\Pr(X = 0) > 0, \qquad \text{and} \qquad \Pr(Y = 0) > 0.$$
If it makes any difference, we can assume they're fully discrete. I'm trying to bound the covariance between $X$ and $Y$ using a mixing coefficient.
A straightforward such bound is:
$$ \operatorname{Cov}(X,Y) \leq 4 \alpha(\sigma(X), \sigma(Y)) \|X\|_\infty\|Y\|_\infty,$$
where $\alpha(\sigma(X), \sigma(Y))$ is the $\alpha$-mixing coefficient between the $\sigma$-algebras generated by $X$ and $Y$, and $\|A\|_\infty$ is the essential supremum of $A$. (See, e.g., Lemma 3 on page 10 in Doukhan, 1994).
However, this does not exploit that we know that there are point masses at zero. Can we use that information to sharpen the bound?
Doukhan (1994), "Mixing: Properties and Examples." Paywall link. Non-paywall link.
For a random variable $Z$ and $u\in[0,1]$, define $$Q_Z\left(u\right):=\inf\left\{t \mid \mathbb P\left(\left|Z\right|\gt t\right)\leqslant u\right\}. $$ Rio's covariance inequality [1] states that for two random variables $X$ and $Y$, $$\left|\operatorname{Cov}\left(X,Y\right)\right|\leqslant 2\int_0^{\alpha\left(X,Y\right)} Q_X(u)Q_Y(u)\mathrm du. $$ In your context, observe that for $t\gt 0$, $\mathbb P\left(|Z|\gt t\right)\leqslant 1-\mathbb P\left(Z=0\right)$ hence $Q_Z\left(u\right)=0$ if $u\geqslant 1-\mathbb P\left(Z=0\right)$. Moreover, $Q_Z(u)\leqslant \lVert Z\rVert_\infty$, hence $$\left|\operatorname{Cov}\left(X,Y\right)\right|\leqslant 2\alpha (X,Y) \min\left\{1-\mathbb P\left(X=0\right),1-\mathbb P\left(X=0\right)\right\}\lVert X\rVert_\infty\lVert Y\rVert_\infty.$$
[1] Rio, Emmanuel. "Covariance inequalities for strongly mixing processes." Annales de l'I.H.P. Probabilités et statistiques 29.4 (1993): 587-597. http://eudml.org/doc/77471.