Expectation of odd one-to-one transformation of a random variable.

Question

Let $X,Y$ be random variables taking values in $[-1,1]$. Suppose we know that $\mathbb{E}[X] = c \cdot \mathbb{E}[Y]$ where $c \in \mathbb{R}^-$, and that neither expectation is $0$. Suppose additionally that for any odd one-to-one function $f : \mathbb{R} \rightarrow \mathbb{R}$, $\mathbb{E}[f(X)] = c \cdot \mathbb{E}[f(Y)]$. What more can we say about the distribution of $X,Y$?

Answer 1

Since $|X|\leqslant 1$, $|Y|\leqslant 1$ a.s. we have $X,Y\in L^p$ for $1\leqslant p\leqslant\infty$. Now, $x\mapsto x^m$ is an odd, injective function for positive odd integers $m$, so $$\mathbb E\left[X^{2n+1}\right] = c\ \mathbb E\left[Y^{2n+1}\right] $$ for all $n\geqslant0$. Further, $\|X\|_{\infty}\leqslant 1$ and $\|Y\|_{\infty}\leqslant 1$. (Not sure if this is helpful, just throwing out ideas.)