I came across the following problem:
Given two functions u,v mapping $[0.5,1]\to (0,1]$ where u(x) is the identity function, $u(x)=x$. Suppose that $\frac{u(x)}{u(1-x)}> \frac{v(x)}{v(1-x)}$ holds. What can we say about the function $v(x)$?
Candidates I was thinking about are concave, log-concave, or other conditions on the derivative.
Any pointers would be appreciated!