Suppose we have a random variables $V$ such that \begin{align} E[V^n]=0 \end{align} for some $n>1$ odd.
My question is what can we conclude about the first moment or other lower moments? For example, can we give meaningful lower and upper bounds.
For example, we can not say that $E[V]=0$. A counter example, would be \begin{align} X=[-2, 1], P_X=[1/9, 8/9] \end{align}
Another thing we can not use is Jensen's inequality since $v^n$ is neither convex or concave for $n$=odd.
Note, that things are much easier with even moments. Since, by Jensen's inequality \begin{align} \left(E[V^{\frac{n}{q}}] \right)^{q} \le E[V^n] \end{align} for any $1 \le q \le n$.
So, if one of the even moments is zero than $V=0$ a.s.
Thank you.