Take nonnegative random variables $X$ whose first $K$ moments have bounds:
$\mu^k\leq E[X^k]\leq c\mu^k$ for each $k=1,\dots,K$.
In this case what is an upper bound for $P(X\leq O(\mu))$?
I am aware of a paper[1] that states the result that the least upper bound is given by the coefficients, but the formula given is very complicated and depends on the exact moments.
[1]: Bounds on a Distribution Function when its First n Moments are Given