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?



A moment problem
The argument from the paper Carlo linked to (which, by the way, is essentially a classical result of Cramer's) can be adapted to your situation.


Consider the moment generating function $f(z) = Ee^{z\ln X}$ of the random variable $\ln X$. By your assumption, this is well defined for $0\le \textrm{Re}\, z<1/2$. Moreover, $f$ is continuous on this strip and holomorphic on the open strip. Therefore, if two such random variables $\ln X$, $\ln Y$ satisfy $f\_X(z)=f\_Y(z)$ for real $0\le z<1/2$, then the $f$'s also agree on $z=ix$, $x\in\mathbb R$. Since the Fourier transform determines the distribution, this gives the claim.


Of course, this argument works for any $c>0$ instead of $1/2$, so it establishes a slightly stronger statement.


