# Upper bounding the start of a distribution's CDF, given bounds on first moments

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.