Show that $E[X^2])\ge \frac{4}{3} (E[X])^2$ for $X$ is supported on $(0,\infty)$ and has a monotone decreasing pdf

Question

I have observed the following inequality numerically:

Suppose that random variable $X$ is supported on $(0,\infty)$ and has a monotone decreasing pdf then $$E[X^2]\ge \frac{4}{3} (E[X])^2$$.

Question: How would one prove this inequality? This is clearly a stronger version of Jensen's or Cauchy-Schwartz which say that $E[X^2]\ge (E[X])^2$.

Also, I am assuming something like this should be known. If so, does this inequality have a name?