The coefficient of multiple determination is an important random variable in statistics. The UMP property depends on the montonicity of the likelihood ratio. The distribution was derived by Fisher, but is not widely used as it involved the hypergeometric function.
The monotonicity of the likelihood ratio comes down to proving that the function:
$\quad\quad\quad\quad g(x) = \frac{\Large _2F_1(a,\,b\,;\,c\,;\,\alpha x)}{\Large _2F_1(a,\,b\,;\,c\,;\,x)},\;\; 0<\alpha<1,\;\,a,b,c>0$
is decreasing on $[0,1]$.
Equivalently, let:
$\quad\quad\quad\quad f(x) = \frac{\Large _2F_1(a,\,b\,;\,c\,;\,x)}{\Large _2F_1(a+1,\,b+1\,;\,c+1\,;\,x)},\;\;a,b,c>0$
Prove that:
$\quad\quad\quad\quad f(\alpha x) > \alpha f(x),\;\;0<\alpha<1,\;0\le x \le 1 $
I.e. $f(x)$ is a "star-shaped" function, and $f(x)/x$ is decreasing.
Equivalently, prove that:
$\quad\quad\quad\quad h(x) = \log\big( {}_2F_1(a,\,b\,;\,c\,;\,x^2)\big),\;\;a,b,c>0$
is convex on $[0,1)$.
Using the series representation of the hypergeometric function, we see that $_2F_1(a,b\,;c\,;x^2)$, $0<x<1$ completely monotonic.
Since completely monotonic functions are log convex, it follows that $\log \big({}_2F_1(a,b\,;c\,;x^2\big)$ is convex on $[0,1)$.