Let $X$ be a random variable defined over $0-1$ and let $X_1 . . . X_4$ be the order statistics of $4$ draws from $X$ (where $X_1$ is the minimum).
I am looking to see if I can identify any general conditions for $X$ such that $\mathbb{E(}X_4 - X_3) < 3\mathbb{E}(X_2 -X_1)$
Intuitively, the validity of the inequality seems like it would be a property of the skewness of $X$. For example, in response to this question (where the inequality in the first part of that question reduces to the one presented here) Sasha showed that the inequality holds for the Beta distribution(a,2) when $a \ > 0.512761$. As $\beta$ increases, I believe the critical value of $a$ increases (for example, when $\beta = 3$, $a > .6017$) but I believe in all cases the distribution has to be extremely positively skewed for it not to hold. However, skewness alone can't be the only factor, as $Skew(Be(.512761,2)) = 1.21989$ and $Skew(Be(.6017,3)) = 1.3617$.
Does anyone have any ideas? I am looking for some way to say "the inequality holds so long as $X$ has _________ properties."