Let $X,Y$ be i.i.d. random variables, $E[X^4]=1$, what's the best upper bound for $E[(X-Y)^4]$ ?

1. A trivial upper bound is $16$, since $(X-Y)^4 \leq 8 （X^4+Y^4)$ then take expectation on both sides. However, equality cannot be achieved.

2. My guess of the best upper bound will be $8$, achieved when $X$ is uniform at random from $\{-1, +1\}$.

Let $X,Y$ be i.i.d. random variables, $\mathbb{E}[X^4]=1$, what's the best upper bound for $\mathbb{E}[(X-Y)^4]$ ?

1. A trivial upper bound is $16$, since $(X-Y)^4 \leq 8 (X^4+Y^4)$ then take expectation on both sides. However, equality cannot be achieved.

2. My guess of the best upper bound will be $8$, achieved when $X$ is uniform at random from $\{-1, +1\}$.