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


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