Is there any direct relation among first, third, fourth order moments?

Question

Suppose that $X$ is a real-valued random variable, with $EX^4=1$ and $EX\leq0$. Find an explicit constant $c<1$ such that $EX^3<c$.

Answer 1

$$ E(X_+) + E(X_-) = E(|X|) \le E(X^4)^{1/4} \le 1 $$ $$ E(X_+) \le E(X_-) $$ Therefore $$ E(X_+) \le 1/2 $$ Apply Hoelder $E(UV) \le E(U^3)^{1/3} E(V^{3/2})^{2/3}$ with $U = X_+^{1/3}$, $V = X_+^{8/3}$: $$ E(X_+^3) \le E(X_+)^{1/3} E(X_+^4)^{2/3} \le (1/2)^{1/3} $$ $$ E(X_-^3) \ge 0 $$ $$ E(X^3) = E(X_+^3) - E(X_-^3) \le (1/2)^{1/3} $$