I am interested in maximal inequalities in the following setup: Let $\left( (X_{n,i})_{i=1,\ldots, n}\right)_{n\in\mathbb N}$ be random variables such that $X_{n,i}$ takes values in $\{0,\ldots, n\}$ for $i=1,\ldots, n$ and such that for all $n$ $$\sum_{i=1}^n X_{n,i}=n.$$ In particular assume that for all $n$ the family $(X_{n,i})_{i=1,\ldots, n}$ is negatively correlated (or even negatively associated) and that for all $a\in \{0,\ldots, n\}$ and $i=1,\ldots, n$ $$\mathbf P(X_{n,1} \geq a) \geq \mathbf P(X_{n,i}\geq a).$$
So it's easy to show (standard result I would say) for $s\geq 0$ $$ e^{ s \mathbf E[M_n]} \leq n \mathbb E\left[ e^{ s X_{n,1}} \right] .$$
But is there anything we can do to get a bound with $\mathbb E\left[{ X_{n,1}} \right]$ on the right side?
I would be glad for any hint, advice or literature...
Wish you all a nice day!