Let $X,Y,Z$ be random variables such that

• $X,Y,Z$ have mean $0$ and variance $1$.
• $X$ and $Y$ are pairwise independent.
• $Z$ can be arbitrarily dependent on $X$ and $Y$.

My question is: What can we say about the dependence of $Z$ on $X$ and $Y$? For example, if $\text{Cov}(X,Z)=1$ (so they are perfectly correlated), then $Z$ and $Y$ must be independent. In general, can we say anything about how much $Z$ can depend on (or be correlated with) $X$ and $Y$? Maybe something like $|\text{Cov}(X,Z)+\text{Cov}(Y,Z)|\leq 1$ holds?