Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. It is also known that the first moment exists for each of them, but a priori one does not know if the first moments are equal.

(It is also known that all negative moments exist, but if possible this assumption should be avoided.)

Is it possible to show that the two random variables have equal distribution?