I have two cdfs, both distributed over 0 to 1. Let's call them $F(x)$ and $G(x)$.
If I know that $\int_0^1 F(x) \,dx < \int_0^1 G(x) \,dx$
then, does it follow that
$ \left|\frac{d}{dn} \int_0^1 F(x)^n \,dx\right| > \left|\frac{d}{dn}\int_0^1 G(x)^n \,dx\right|$ where $n$ is a positive real number >1? (I used to say integer but corrected due to comments)
I would think this would be the case, because the exponent will have a greater effect on a smaller fraction, and $F$ has to have smaller fractions on average. But, generally whenever I say, "I would think this would be the case" I am wrong and not thinking of something.... so I pose the question to the awesome SE community...