Suppose $X$ and $Y$ are two random variables where $X$ has SOSD (second order stochastic dominance) over $Y$. Let $g(\cdot)$ be a monotonic function and $X' = g(X)$ and $Y' = g(Y)$.
Under what conditions of $g(\cdot)$ $X'$ has SOSD over $Y'$? I know that if $g$ is linear, SOSD property is reserved. Is there any sufficient and necessary conditions of $g$ that assures SOSD property?