### Problem statement

Let $P$ be a probability measure on the positive real line and assume all it's *raw moments*, $\mu_k = \mathbb{E}[x^k]$, $k=1,2,\dots$ exist and $\mu_k < \infty$ for all $k$. Let $\mu = \mu_1$ be the mean of $P$ and assume $\mu > 0$.

Does the following inequality hold? $$ 4 \frac{\mu_3}{\mu^3} + 6 \frac{\mu_2}{\mu^2} - 9\left(\frac{\mu_2}{\mu^2}\right)^2 - 1 \leq 0 $$

### Further remarks

I have tried to find a counterexample experimentally and found the above holds over a wide variety of continuous distributions (i.e. Gamma and Beta distributions) and may be tight in the sense that there are distributions which are just below zero.

### Pointers

From *Stieltjes moment problem* we know that for any measure on the positive real line we must have that

$$ \det\left(\left[\begin{array}{cc} 1 & \mu\\ \mu & \mu_2 \end{array} \right]\right) > 0, \qquad\textrm{and}\qquad \det\left(\left[\begin{array}{cc} \mu & \mu_2\\ \mu_2 & \mu_3 \end{array}\right]\right) > 0, $$ from which we can infer that $$ \frac{\mu_2}{\mu^2} > 1, \qquad\textrm{and}\qquad \frac{\mu_3}{\mu^3} > \left(\frac{\mu_2}{\mu^2}\right)^2. $$

(But I have not been able to show the above inequality based on this.)