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 $P(X\leq O(\mu))$?




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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](https://projecteuclid.org/euclid.aoms/1177728977) 



Upper bounding the start of a distribution's CDF, given bounds on first moments
Have a look at:


[Moment information for probability distributions, without solving the moment problem, II: Main-mass, tails and shape approximation](https://www.sciencedirect.com/science/article/pii/S037704270800513X)


P.N.Gavriliadis et al., Journal of Computational and Applied Mathematics 229(1), 2009


