Expectation of concave transformation of random variable

Suppose I have two different discrete random variables $y>0$ and $x>0$. Now I want to compare two expected values involving these and a nonlinear transformation: When is one larger than the other, i.e., when is $$E\left[y^\alpha\right]>E\left[x^\alpha\right]?$$ We can assume $0<\alpha<1$ (making the transformation concave). Now what condition on the distributions of these random variables are necessary or sufficient for this inequality to hold?

Two cases that have a simple answer:

1. For $\alpha=1$ there is no nonlinearity, so the matter is quite easy and simply boils down to which random variable has a larger mean.

2. For another nonlinear transformation of the random variables, $E[y-y^2]$ and $E[x-x^2]$, the question can be reduced to a sum of mean and variance of the random variables, since $E[y-y^2]=E[y]-(Var(y)+E[y]^2)$.

Since the above transformation $E\left[y^\alpha\right]$, $0<\alpha<1$ is concave, there should be some condition involving some measure of mean and variance (and possibly higher order moments) as well. However, within the class of concave transformations, the quadratic case seems to be unique in that it gives such a simple condition on the distributions (only involving the mean and variance of the random variables). Or is there a similarly simple condition?

(I am adding the tag "economics" since the problem is similar to an expected utility problem comparing two gambles.)
