The following post is related to If $x \in [0,1]$ then how do you show the inequality $r > s > 0$ implies $x^r < x^s$? and can be thought of as a generalization to the questions posted previously. The backround for these problems comes from the need to rule out cases of fractional powers when the inequality below is reversed in (particular $x^s > y^r$). The question can be formulated as follows
If $ 0< x < y$ and $0 < s < r$ then when is the following inequality true $x^s < y^r$?