Can someone help prove the following upper bounds: Note: First inequality has absolute values.
(1) $E |X_i X_j X_k| \leq \frac{1}{3} (E|X_i^3| + E|X_j^3| + E|X_k^3|) $
(2) $E[X_i X_s] E[X_j X_t] \leq \frac{1}{4} (E[X_i^4] + E[X_s^4] + E[X_j^4] + E[X_t^4]) $
Answer for (1): as shown by multiple methods in Proving an inequality with convexity
we have $abc \leq \frac 1 3(a^{3}+b^{3}+c^{3})$ for $a,b,c \geq 0$. Just put $a=|X_i|, b=|X_j|, c=|X_k|$.
Hint for 2): Just apply Holder/Cauchy-Schwarz inequality twice for $EX_iX_s$ and $EX_jX_t$ and the apply AM-GM inequality. [ Note that $EX_iX_t\leq \sqrt {EX_i^{2}} \sqrt {EX_t^{2}} \leq (EX_i^{4})^{1/4} (EX_t^{4})^{1/4}$].